FIXING [OUR] EYES ON JESUS: aphorontes (PAPMPN) eis ton tes pisteos archegon kai teleioten Iesoun: (Hebrews 12:3)

See the well written devotional Fix Your Eyes On Jesus by Ann Ortlund)

"looking with undivided attention" (Analyzed Literal)
"Looking away [from all that will distract] to Jesus" (Amplified)
"looking steadfastly on Jesus" (Darby)
"simply fixing our gaze upon Jesus" (Weymouth)
"looking off and away to Jesus" (Wuest)

Westcott comments that...

The encouragement to be drawn from earthly witnesses passes into the supreme encouragement which springs from the contemplation of Christ. Above the ‘cloud of witnesses,’ who encompass us, is our King, no Roman Emperor dispensing by his arbitrary will life or death to the stricken combatant, but One Who has Himself sustained the struggle which we bear. He Who is ‘the captain (author) of our salvation,’ ‘the righteous Judge’ (see note 2 Timothy 4:8), is also the example and the inspiration of our faith. He in His humanity endured suffering and shame beyond all others and received compensating joy and glory. We therefore may hope by sharing His sufferings to share His glory (see note Romans 8:17) (The Epistle to the Hebrews)

Fixing our eyes (872) (aphorao from apo = away from something near + horao = look, see, behold) means to look away from other things and steadfastly or intently toward a distant object. The idea is to  direct one’s attention without distraction

The only other NT use is

Philippians 2:23 (note) Therefore I hope to send him immediately, as soon as I see how things go with me (Comment: Here the connotation is that of learning or finding out about another. There is one use in the Septuagint of Jonah 4:5 which carries a similar meaning to this use in Philippians.)

Our eyes gaze at that which engrosses our heart. If worldly and fleshly desires are harbored in our heart, they will obscure our moral vision and our ability to see Jesus as we should. Keeping our eyes fixed on Jesus is the only safe way to live in a sinful world. This involves looking "away from all else" that would turn us aside.

Westcott writes that they are to be...

looking away from all that distracts on earth into...not only at the first moment, but constantly during the whole struggle...Christ is always near and in sight... In one form or other the hope of the vision of God has been the support of the saints in all ages (Ibid)

Kenneth Wuest writes that aphorao...

means “to turn one’s mind to a certain thing.” Both meanings are applicable here, the spiritual vision turned away from all else and together with the mind, concentrated on Jesus. What a lesson in Christian running technique we have in that little preposition “off, away from,” which is prefixed to this verb. The minute the Greek runner in the stadium takes his attention away from the race course and the goal to which he is speeding, and turns it upon the onlooking crowds, his speed is slackened. It is so with the Christian. The minute he takes his eyes off of the Lord Jesus, and turns them upon others, his pace in the Christian life is slackened, and his onward progress in grace hindered. Messiah is called the author of our faith. (Wuest, K. S. Wuest's Word Studies from the Greek New Testament: Eerdmans or Logos)

The present tense describes a habitual attitude (which is necessary to run with endurance) and not just a single act. Aphorao conveys the idea of "having eyes for no one but Jesus." It is good to "glance" at the godly examples of the cloud of Old Testament witnesses and be encouraged by their faithful finish, but it is imperative that each of us then firmly fix our gaze on our Savior Jesus. The idea is that we looking away from everything which may distract. The idea is to "have eyes for no one but Jesus."

Expositor's adds that...

We are to run this race "with no eyes for any one or anything except Jesus" (Moffatt, in loc.). It is He toward Whom we run. There must be no divided attention. (Gaebelein, F, Editor: Expositor's Bible Commentary 6-Volume New Testament. Zondervan Publishing)

Am I continually fixing my eyes on Jesus?
Am I looking away from the problems and concentrating on Jesus ?

Keep your eyes
On the prize.

The Greek verb aphorao pictures a runner who is virtually oblivious to the thousands of onlookers, even as his attention is diverted from every consideration except that of running the best possible race. We've all seen runners and other top athletes who are "practicing focusing" prior to the race or event. Their attention is concentrated upon one thing to the total exclusion of everything else. And so the runner looks away from everything else and fixes his or her eyes upon the goal and not the cheering crowds or even the opponents. In a race if the runner turns his or her head even slightly toward the spectators or the opponents, their speed will be lessened. 

Peter learned the lesson of not keeping his gaze on Jesus in Matthew 14...

26 And when the disciples saw Him walking on the sea, they were frightened, saying, "It is a ghost!" And they cried out for fear.
27 But immediately Jesus spoke to them, saying, "Take courage, it is I; do not be afraid."
28 And Peter answered Him and said, "Lord, if it is You, command me to come to You on the water."
29 And He said, "Come!" And Peter got out of the boat, and walked on the water and came toward Jesus.
30 But seeing the wind, he became afraid, and beginning to sink, he cried out, saying, "Lord, save me!"
31 And immediately Jesus stretched out His hand and took hold of him, and said to him, "O you of little faith, why did you doubt?"
32 And when they got into the boat, the wind stopped.

Some storms come because of our disobedience, but this one came because the disciples obeyed Jesus. Peter obeyed Jesus' command and walked on the water; but became distracted by the waves, and took his eyes off Jesus. We look to Jesus by faith when we trust His word. We need to beware of the distracting storms (which are certainty in our life in this fallen world)! Instead we need to keep our eyes continually focused on the One Who controls the "waves" and is able to calm the storm!

Robert Murray McCheyne (Biography) who died at in the thirtieth year of his age and seventh of his ministry" had some sage advice for his age...

For every one look at your problems, your weaknesses, your failures -- take ten looks at Jesus. (Mark it down - Worry will keep you from fixing your eyes on Jesus, or fixing your eyes on Jesus will keep you from worry).

Warren Wiersbe reminds us of the importance of a "look" writing that...

 It was in “looking to Him” that we were saved, for to look means “to trust.” When the dying Jews looked to the uplifted serpent, they were healed; and this is an illustration of our salvation through faith in Christ (Numbers 21:4-9; John 3:14-16). “Looking unto Jesus” describes an attitude of faith and not just a single act." (Wiersbe, W: Bible Exposition Commentary. 1989. Victor or Logos) (Bolding added)

I See Jesus

I don't look back: God knows the fruitless efforts,
The wasted hours the sinning, the regrets;
I leave them all with Him Who blots the record,
And mercifully forgives, and then forgets

I don't look forward, God sees all the future,
The road that, short or long, will lead me home,
And He will face with me its every trial,
And bear for me the burdens that may come.

I don't look round me: then would fears assail me,
So wild the tumult of earth's restless seas;
So dark the world, so filled with woe and evil,
So vain the hope of comfort or of ease.

I don't look in; for then am I most wretched;
Myself has naught on which to stay my trust;
Nothing I see save failures and short-comings,
And weak endeavors crumbling into dust.

But I look up -- into the face of Jesus,
For there my heart can rest, my fears are stilled.
And there is joy, and love, and light for darkness,
And perfect peace, and every hope fulfilled
--Annie Johnson Flint

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Guy King says that there's a gold running cup on another man's mantel that could have been -- should have been -- on his own.  He was running toward the tape, coming in number one.  Somebody was trailing on his right, and he shot a look to see where he was. It was a split-second distraction that his competitor needed, and he flashed by him and won.  "Our sole safety [as Christians]," says Guy King, "is to be found in keeping our eyes averted . . . from others, and keeping them unswervingly 'looking unto Jesus.'" (Brought In)

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A famous illustration emphasizes this basic principle of running the race with focus:

On May 6, 1954, Roger Bannister became the first man in history to run a mile in less than 4 minutes. Within 2 months, John Landy eclipsed the record by 1.4 seconds. On August 7, 1954, the two met together for a historic race. As they moved into the last lap, Landy held the lead. It looked as if he would win, but as he neared the finish he was haunted by the question, "Where is Bannister?" As he turned to look, Bannister took the lead. Landy later told a Time magazine reporter, "If I hadn't looked back, I would have won!"

One of the most descriptive pictures of the Christian life in the Bible is of an athlete competing in a race. First Corinthians 9:24-27 tells us that discipline is the key to winning. In Hebrews 12:1-2, we are encouraged to lay aside anything that might hinder our spiritual advancement and to stay focused on Christ. And in Philippians 3:12-13, the apostle Paul said, "I press on, . . . forgetting those things which are behind and reaching forward to those things which are ahead."

Lord, give us endurance as we run this race of life. Help us not to wallow in past failures, but to be disciplined and to shun sinful ways. May we fix our eyes on the eternal goal set before us and keep looking unto Jesus. --H G Bosch (Our Daily Bread)

Run the straight race through God's good grace,
Lift up thine eyes and seek His face;
Life with its way before us lies,
Christ is the path and Christ the prize. --Monsell

You can't make spiritual progress
by looking back.

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What a lesson for the Christian. The minute we turn our eyes toward our fellow-men and take them off the Lord Jesus, our pace is slackened. Pride, discouragement, envy, the desire for praise, these and other evils incapacitate the Christian runner as he looks at men instead of keeping his eyes fixed upon Jesus.

Turn Your Eyes Upon Jesus
by Helen Lemmel

 O soul, are you weary and troubled?
No light in the darkness you see?
There’s light for a look at the Savior,
And life more abundant and free!

Refrain
Turn your eyes upon Jesus,
Look full in His wonderful face,
And the things of earth will grow strangely dim,
In the light of His glory and grace.

Through death into life everlasting
He passed, and we follow Him there;
O’er us sin no more hath dominion—
For more than conquerors we are!
Refrain

His Word shall not fail you—He promised;
Believe Him, and all will be well:
Then go to a world that is dying,
His perfect salvation to tell! 
Refrain
 

It is possible that some of the Jewish Christian readers had begun to take their eyes off their Messiah and to fix them instead on the hardships they were encountering for the sake of Christ.

Don't look at your fellow Christian stumbling along. Don't look at any great Christian that you admire. He might take you in a different path. Note well that the eyes are very important in any athletic endeavor. Looking unto Jesus - that is our occupation!

Keep your eyes on the Lord
He never takes His eyes off you.

A W Pink has an excellent note on "fixing our eyes on Jesus" writing that...

The person of the Saviour is to be the “mark” (Ed note: the Greek word for "mark" is skopos =  that on which the eye is fixed, the distant mark looked at, the goal or end that one has in view) on which the eyes of those who are pressing forward for the prize of the high calling of God, are to be fixed. Be constantly “looking” to Him, trustfully, submissively, hopefully, expectantly. He is the Fountain of all grace (John 1:16): our every need is supplied by God “according to His riches in glory by Christ Jesus” (see note Philippians 4:19). Then seek the help of the Holy Spirit that the eye of faith be steadfastly fixed on Christ. He has declared “I will never leave thee, nor forsake thee,” then let us add, “The Lord is my Helper, I will not fear what man shall do unto me” (see note Hebrews 13:5; 13:6). Salvation is by grace, through faith: it is through “faith” we are saved, not only from Hell, but also from this world (1 John 5:4), from temptation, from the power of indwelling sin—by coming to Christ, trusting in Him, drawing from Him.

What are the things which hinder us running? An active Devil, an evil world, indwelling sin, mysterious trials, fierce opposition, afflictions which almost make us doubt the love of the Father. Then call to mind the “great cloud of witnesses”: they were men of like passions with us, they encountered the same difficulties and discouragements, they met with the same hindrances and obstacles. But they ran “with patience,” they overcame, they won the victor’s crown. How? By “looking unto Jesus”: see note Hebrews 11:26. But more: look away from difficulties (see note Romans 4:19), from self, from fellow-racers, unto Him who has left us an example to follow, in whom dwelleth all the fulness of the Godhead bodily, so that He is able to succour the tempted, strengthen the weak, guide the perplexed, supply our every need. Let the heart be centered in and the mind stayed upon Him.

The more we are “looking unto Jesus” the easier will it be to “lay aside every weight.” It is at this point so many fail. If the Christian denies self of different things without an adequate motive (for Christ’s sake), he will still secretly hanker after the things relinquished, or ere long return to them, or become proud of his little sacrifices and become self-righteous. The most effective way of getting a child to drop any dirty or injurious object, is to proffer him something better. The best way to make a tired horse move more quickly, is not to use the whip, but to turn his head toward home! So, if our hearts be occupied with the sacrificial love of Christ for us, we shall be “constrained” thereby to drop all that which displeases Him; and the more we dwell upon the joy set before us, the more strength shall we have to run “with patience the race that is set before us.” (Pink, A. W: An Exposition of Hebrews) (Bolding added)

Jesus (2424) (Iesous equivalent to the Hebrew Yeshu'a H3091 = Jehovah His help, Jehovah is salvation or Jehovah delivers)  means He saves, helps or delivers. Vine says Jesus is a transliteration of the Hebrew name “Joshua” which means "Jehovah is salvation".

Vincent writes...

Having presented a long catalogue of witnesses under the old covenant, he now presents Jesus, the mediator of the new covenant, and the supreme witness. (Vincent, M. R. Word Studies in the New Testament 4:538)

Writing to Timothy Paul refers to the "witness" of Jesus...

I charge you in the presence of God, Who gives life to all things, and of Christ Jesus, Who testified (martureo - witnessed, gave evidence) the good confession before Pontius Pilate

John refers to Jesus twice as a "witness"..

Jesus Christ, the faithful witness, the first-born of the dead, and the ruler of the kings of the earth. To Him who loves us, and released us from our sins by His blood, (see note Revelation 1:5)

(Jesus speaking) And to the angel of the church in Laodicea write: The Amen, the faithful and true Witness, the Beginning of the creation of God, says this... (see note Revelation 3:14)

We do well to fix our eyes on the One Whose testimony was and is both faithful (worthy of our trust) and true (containing no deceit or lies, etc.) Whenever you are tempted to look at your circumstances or at yourself, look to Jesus by faith and rejoice in His faithfulness.

John Piper reminds us that...

Jesus is the one sure thing. He is the rock (Ed note: Click for Scripture chain & chart - studying these passages observing what they teach about Jesus as the Rock would make a great Sunday School lesson), the foundation. Get Him in view when everything else in your life is swirling and you will land safely. The coach knows that panic is the worst enemy in the midst of a dangerous routine. And panic comes from a building sense of weakness and uncertainty—or we might say, a lack of assurance in the Christian life. And the main cry of this book is: Jesus is sure! God is sure! The Cross is sure! The Covenant is sure! The promises are sure! Fix your attention on the things that are sure! “Find the floor!” (Read his full message The Doctrine of Perseverance: The Earnest Pursuit of Assurance )

In another message Piper says that the idea of looking to Jesus means that you...

do not dwell on yourself, dwell on what God has done in Jesus Christ. There is a paradox here. For many people—most people, I think—the more we focus on the subjective inner workings of our own soul and the relative purity or impurity of our own attitudes and behavior, the more uncertain we become of our own assessment of our authenticity. Paradoxically the path to assurance is to shift our focus off of ourselves and onto God. Off of the subjective and onto the objective. (Read his full message God Has Chosen Us in Him Before the Foundation of the Earth)

Spurgeon has a similar thought writing...

How frequently you who are coming to Christ look to yourselves. "Oh," say you, "I do not repent enough!" That is looking to yourself. "I do not believe enough!" That is looking to yourself. "I am too unworthy." That is looking to yourself. "I cannot discover," says another, "that I have any righteousness." It is quite right to say that you have not any righteousness, but it is quite wrong to look for any....Look to Him, look at Him, study Him, know all you can about Him, meditate upon Him.

Piper observes that this passage makes the point...

that the fight of faith is not done in our own strength. When you go away to plan your fall run with Jesus, verse 2 says, “Look to Jesus the author and perfecter of your faith.” Don’t look to your own resources and say, “I’ve tried before. It won’t work.” Fix your eyes on him. The battle is a battle of faith: will you believe that the things he promises are better than the bad habits that you use to cover your sadness? (Read the full message Running with the Witnesses)

Just as a runner concentrates on the finish line, believers should concentrate on Jesus, the goal and objective of our faith. Paul wrote

"Brethren, I do not regard myself as having laid hold of (taken eagerly; seized, possessed) it yet; but one thing (singleness of purpose; focused) [I do = not in the original Greek]: forgetting (completely forgetting; present tense = continually forgetting) what lies behind and reaching forward (attempting energetically to attain the goal like a runner stretching his head out to cross the tape the winner; present tense = lifelong attitude) to what lies ahead (in front of), I press on (follow hard after, pursuing with desire, earnestness and diligence in order to obtain ~ "zeal" = burning desire to please God, to do His will & to advance His glory) toward the goal for the prize of the upward call of God in Christ Jesus." (see notes Philippians 3:13; 14)

Bruce writes that we are to keep our eyes riveted on Jesus, the foremost Runner for only One...

...stands out conspicuous above all the rest ... the Man who first perfectly realised the idea of living by faith ... , who undauntedly endured the bitter suffering of the cross, and despised the ignominy of it, sustained by a faith that so vividly realised coming joy and glory as to obliterate the consciousness of present pain and shame. (Bruce, A B: Hebrews, P 415-416)

MacDonald adds that Jesus...

"not only began the race but finished it triumphantly. For Him the race course stretched from heaven to Bethlehem, then on to Gethsemane and Calvary, then out of the tomb and back to heaven. At no time did He falter or turn back. He kept His eyes fixed on the coming glory when all the redeemed would be gathered with Him eternally. This enabled Him to think nothing of shame and to endure suffering and death. (MacDonald, W & Farstad, A. Believer's Bible Commentary: Thomas Nelson  or Logos

Vincent writes that the author...

Having presented a long catalogue of witnesses under the old covenant, now presents Jesus, the Mediator of the new covenant, and the supreme Witness. (Vincent, M. R. Word studies in the New Testament. Vol. 4, Page 538)

A W Pink agrees writing that...

“The cloud of witnesses is not the object on which our heart is fixed. They testify of faith, and we cherish their memory with gratitude, and walk with a firmer step because of the music of their lives. Our eye, however, is fixed, not on many, but on One; not on the army, but the Leader; not on the servants, but the Lord. We see Jesus only, and from Him we derive our true strength, even as He is our light of life” (A. Saphir). In all things Christ has the pre-eminence: He is placed here not among the other “racers,” but as One who, instead of exemplifying certain characteristics of faith, as they did, is the “Author and Finisher” of faith in His own person." (The Object of Faith)

Keep choosing to turn your eyes away from the trials and sorrows that are nearby (also from the great conflict of sufferings that was probably imminently looming on the horizon in 70AD as well as the daily reproaches from other Jews calling them to turn away from Messiah).

A T Robertson says

Fix your eyes on Jesus, after a glance at “the cloud of witnesses” (Hebrews 11) for He is the Goal." (Robertson, A T: Word Pictures in the New Testament)

Moses ran his "race". looking with eyes of faith to Him Who is unseen:

By faith Moses, when he had grown up, refused to be called the son of Pharaoh’s daughter;25 choosing rather to endure ill-treatment with the people of God, than to enjoy the passing pleasures of sin;26 considering the reproach of Christ greater riches than the treasures of Egypt; for he was looking to the reward. 27 By faith he left Egypt, not fearing the wrath of the king; for he endured, as seeing Him who is unseen. (see notes Hebrews 11:24;11:25; 11:26; 11:27)

Looking to Jesus...

Why look?  Because the best beings in the universe encourage it (the Angels, the redeemed in heaven, the Holiest on earth), our own needs demand it (we need a Mediator, Example, Friend, such as He is) and finally because the great God enjoins it.

How to look? By the study of His biography. By communion with others of pure heart.

When to look?  At the beginning of the Christian life. I times of temptation and difficult. In all the encouragements and discouragements of life. At death.

What should be our attitude? Trustful. Obedient.. Loving. (Adapted from the Biblical Illustrator)

C H Spurgeon discusses looking to Jesus...
 

Robert Murray McCheyne advised that...

For every one look at your problems, your weaknesses, your failures -- take ten looks at Jesus.

C. M. Merry writes that...

1. “Looking to Jesus” supplies the strongest motive to run well our Christian race; that is, love towards Himself. You know that fire and force are the effect of a supreme affection; how it makes light of difficulties, and changes leaden feet into feet of angel swiftness. Love lightens toil, and makes even waiting more than endurable.

2. “Looking to Jesus” furnishes all needful strength for running well our Christian race. This is the act on our part that appropriates it for our various occasions and exigencies; just as plants, by opening out their
leaves, to them the organs of assimilation, imbibe the light and dew, and distribute sustenance through their entire structure, so we, by “ looking to Jesus,” receive those communications of a spiritual kind, upon which the life of our souls and the vigor of our Christian walk depend.

3. “Looking to Jesus “ brings before us the highest example of a successful runner in the Christian race. When you are in doubt, ask, “What, in such a case, would my Master have done?” (Biblical Illustrator)

F B Meyer writes that we should

...look off unto Jesus. Away from past failure and success; away from human applause and blame; away from the gold pieces scattered on the path, and the flowers that line either side. Do not look now and again, but acquire the habit of looking always; so that it shall become natural to look up from every piece of daily work, from every room, however small, from every street, however crowded, to His calm face; just as the sojourner on the northern shores of Geneva’s lake is constantly prone to look up from any book or work on which the attention may have been engaged, to behold the splendour and glory of the noble range of snowcapped summits on the further shores. And if it seems hard to acquire this habitual attitude, trust the Holy Spirit to form it in your soul.

Above all, remember that where you tread there your Lord trod once, combating your difficulties and sorrows, though without sin; and ere long you shall be where He is now. Keep your eye fixed, then, on him as he stands to welcome and reward you; and struggle through all, animated by his smile, and attracted to his side, and you will find weights and unbelief dropping off almost insensibly and of themselves.

This is the only way by which souls can be persuaded. Argue with them; urge them; try to force them-and they will cling the closer to the encumbrances which are clogging their steps. But present to them Jesus in the beauty and attractiveness of his person and work, and there will be a natural loosening of impediments; as the snow which had been bending the leaves to the earth drops away when the sun begins to shine. And God never takes aught from us, without giving us something better. He removes the symbol, to give us the reality; breaks the type, to give the substance; releases us from the natural and human, to give us the divine. Oh, trust him, soul: and dare to let go, that thou mayest take; to be stripped, that thou mayest become clothed! (Meyer, F B: Hebrews 12:1-2: Stripping for the race)

G Campbell Morgan sums up this section:

These words catch up and apply all that had been said as to the service ren­dered in the past by those who had "received the promises," and had died, not having "received the promise." If they so endured with courage and cheerfulness, we also should be prepared to endure with patience, and run the race toward the glorious goal without wavering, however hopeless the enterprise may seem, when judged by the circumstances of the hour. The ultimate strength of this appeal, however, lies in the contrast which it suggests between these men of the past and ourselves. They had the promises; we have Jesus. They look for the City; we look off unto Jesus. This means that in Him we have a clearer revelation of the glory of the City, and of the travail through which alone it can be built. Through Him our understanding of what the tabernacling of God with men means, is more perfect. In Him the call is to yet profounder suffering and to greater patience. But He is Himself the File-leader of the Faithful; that is, in His own life and service He takes precedence of all others. And so He is supremely the Vindicator of faith in the promises of God as the one principle which moves toward the fulfilment of those very promises. He also is waiting for the consummation, waiting till His enemies shall be made the footstool of His feet, but waiting in the perfect assurance of the final victory. Then we are called upon to rest in His assurance, to have fellowship with His sufferings, and so to hasten the coming of the Day of God. (Morgan, G. C.: Life Applications from Every Chapter of the Bible). (Bolding added)

J. C. Ryle commenting on "looking to Jesus" has some excellent advice: