AND THOUGH YOU HAVE NOT SEEN HIM: on ouk idontes (AAPMPN): (See excellent cross references on seeing the unseeable - Jn 20:29 2Co 4:18 5:7 Heb 11:1,27 1Jn 4:20) (See Piper's sermon "True Christianity: Inexpressible Joy in the Invisible Christ")

Seen (3708) The Greek phrase is "of Whom not having had a glimpse". 

The emphasis in this phrase is upon the person of Christ – "Whom".  The Greek implies the idea of "toward whom" placing emphasis upon the direction of our faith toward a person

Unlike Peter, these readers were not eyewitnesses of Jesus, but this was not a "spiritual disadvantage." The world says "seeing is believing" but Jesus said

"Blessed are they who did not see, and yet believed." (Jn 20:29)

Another of those glorious "paradoxes" of Christianity.

Paul reminded the Corinthians that

we look (skopeo - spy out, look towards an object, to contemplate, give attention to - so not merely a description of usual human vision but with the idea of regarding a thing as important) not at the things which are seen (these things are not the goal of their existence), but at the things which are not seen; for the things which are seen are temporal, but the things which are not seen are eternal." (2 Co 4:18)

Jowett comments that

to be able to see the first is sight; to be able to see the second is insight. The first mode of vision is natural, the second mode is spiritual. The primary organ in the first discernment is intellect; the primary organ in the second discernment is faith... All through the Scriptures this contrast between sight and insight is being continually presented to us, and everywhere we are taught to measure the meagerness and stinginess of the one, and set it over the fulness and expansiveness of the other." (Jowett, Life in the Heights, pp68, 69)

Peter's readers had fully developed the skill of "spiritual insight" and had learned to "walk by faith, not by sight." (2Cor 5:7)

How do we develop that same "spiritual insight"? By believing what Scripture says and living by that belief.

Michael Card expressed the paradox of not seeing yet seeing in one of his songs...

Peter's readers

never saw the Lord Jesus with the physical sense of sight, but ah, what a vivid portrait of Him did the Holy Spirit paint for them on the canvas of their spiritual vision....The picture of the earthly Lord Jesus in His mortal body, seen by human eyes, is supplanted now by the picture of the glorified Man in the Glory, painted by the Holy Spirit for the spiritual vision of the saint....It is as we free ourselves from the conception an artist may have of what he thinks the Lord Jesus looked like in His life on earth, and depend upon the Holy Spirit through the Word to reveal to us the likeness of our Lord Jesus, that we come to some true conception of Him in His glorified state. We will recognize Him in the Glory over yonder, not by what human artists have conceived Him to be, but by the Holy Spirit’s portrait of Him. (Wuest, K. S. Wuest's Word Studies from the Greek New Testament: Eerdmans or Logos)

 YOU LOVE HIM: agapate (2PPAI): (See Torrey's Topic "Love to Christ")

Love (25)  (agapao - see related study of noun agape) means to love unconditionally and sacrificially as God Himself loves sinful men (John 3:16), the way He loves the Son (John 3:35, 15:9, 17:23, 24).

In short, agapao describes unconditional, sacrificial love which is the love that God Himself is. Agapao is not sentimental or emotional but obedient as an act of one's will which desiring another's highest good (and so is willing to die to the interest of self).

Agapao is present tense, picturing their love as the habit of their lives, not an intermittent passing infatuation or emotional high. And if the habit of their life was to "love" Him, they were also "obeying" Him because Jesus taught that

If you love Me, you will keep My commandments." (Jn 14:15 cf Jn 15:13)

Paul records that

the love (agape) of God has been poured out (lavishly poured out to the point of overflowing - perfect tense = has poured in and still floods our hearts) within our hearts through the Holy Spirit who was given to us." (see note Romans 5:5)

The Holy Spirit has poured out God’s love into our hearts and we return that love to Him.

Wiersbe comments that

When you find yourself in some trial, and you hurt, immediately lift your heart to Christ in true love and worship. Why? Because this will take the poison out of the experience and replace it with healing medicine. Satan wants to use life’s trials to bring out the worst in us, but God wants to bring out the best in us. (Wiersbe, W: Bible Exposition Commentary. 1989. Victor or Logos

It is interesting to note as an aside that the expression those who love Him is often a descriptive name for believers (see note Romans 8:28 1Cor 2:9 Ex 20:6 Ps 97:10 [Spurgeon's comment] 145:20 [Spurgeon's comment]). Their love for Christ was a sure mark that they had been born into the family of God, for as Jesus informed the antagonistic Jewish audience (who initially had expressed an intellectual belief in Him, see Jn 8:30ff)...

If God were your Father, you would love Me (Jn 8:42)

Paul was a bit more direct writing that

If anyone does not love the Lord, let him be accursed. Maranatha. (which means "Come Lord") (1Cor 16:22)

Human love needs the seen presence of the beloved one to complete the fullness of its joy (2Jn 1:12) but Peter's readers love of the unseen Jesus produced joy even amid afflictions unspeakable!

Wuest:

It was the clear-cut conception of the Lord Jesus which the Holy Spirit had given these saints through the Word, that caused them to love Him. The distinctive Greek word for “love” here, agape, refers to a love that is called out of one’s heart by the preciousness of the person loved. But even the preciousness of the Lord Jesus would not have made these individuals love Him if God in salvation had not produced in them that divine love which He Himself is, with which to love Him (see note Romans 5:5; Gal 5:22, 23). One must have the nature of an artist to really appreciate and love art. One must have the nature of God (see note 2 Peter 1:4) to appreciate and love the Lord Jesus. It is this ideal combination of a study of God’s Word and a definite subjection to the Holy Spirit that results in the clear, vivid portrait of the Lord Jesus in the spiritual vision of the saint. To know Him is to love Him. To know Him better, is to love Him better. The secret of an intimate, loving fellowship with the Lord Jesus, the secret of knowing Him in an intimate way, is in the moment-by-moment control of the Holy Spirit over the life of the Christian believer." (Wuest, K. S. Wuest's Word Studies from the Greek New Testament: Eerdmans or Logos)

William Lincoln writes

People talk a lot about love, but the true test of love to God and Christ is, that in the trial it says—“I would not lose the favor and smile of God, so will rather suffer than grieve Him.” Love will be content with a crust and the smile of God, rather than a better position and the popularity of the world without it. Such tests must come to all the true children of God; they winnow the chaff from the wheat. The gold comes out from the fire tried, and purified from its dross."

AND THOUGH YOU DO NOT SEE HIM NOW: eis on arti me orontes (PAPMPN):

This fact points out that their continuing committal is not nurtured by gazing upon His visible presence but by grazing upon the promises in His Word, for faith comes by hearing and hearing by the Word of Christ, Who is "the Word" (Jn 1:1). Unlike Thomas Peter's readers had not demanded to see Him before they would believe (John 20:25).

The Life Application Commentary has this interesting note about our "paradoxical" (to the outside world) faith

Christians rejoice despite trials and suffering, have faith in someone they have never seen, and stake their lives on promises. Why? Because they know the Lord." (cp notes 2 Timothy 1:12) (Barton, B. B., et al. Life Application Bible Commentary. Romans: Tyndale House Publishers or Logos)

Now (737) (arti) means at this moment, at the immediate present, or "just now" and has a beautiful implication in the present context. Peter is saying in essence "No "not now" yet yes "one day soon and forever" you will see your Lord and King face to face." (see notes Revelation 22:3; 22:4)  This great hope (certainty) must have made them want to shout "Hallelujah!" Does this glorious prospect make you want to shout "glory" or do you want to "shrink back in shame at His coming" (see 1Jn 2:28)

BUT BELIEVE IN HIM: pisteuontes (PAPMPN) de: 

Believe (4100)  (pisteuo from pistis; pistos; related studies the faith, the obedience of faith) (used 3x in 1 Peter) denotes not just an intellectual assent to the truth of the gospel, but also involves an act of the heart and will of the subject.

Pisteuo is present tense emphasizes the ongoing commitment (in the face of not seeing Him...walk by faith not by sight).

Pisteuo means  to consider something to be true and therefore worthy of one’s trust.  To accept as true, genuine, or real. To have a firm conviction as to the goodness, efficacy, or ability of something or someone. To consider to be true. To accept the word or evidence of.

means to persuade, to cause belief, to induce one to do something by persuading, and so runs into the meaning of to obey, properly as the result of persuasion

Did you notice the triad of hope, love and faith in (1 Peter 1:3-8)? Faith means surrendering all to God and obeying His Word in spite of circumstances and consequences. Love and faith go together for when you love someone, you trust him. And faith and love together help to strengthen hope for where you find faith and love, you will find confidence for the future. This faith in the Messiah Whom they had never seen sustained them.

W. H. Griffith Thomas gives us an excellent description of the interrelation of faith, hope and love...

YOU GREATLY REJOICE WITH JOY INEXPRESSIBLE AND FULL OF GLORY: de agalliasthe (2PPMI) chara aneklaleto kai dedoxasmene (RPPFSD):  (Torrey has an interesting topic "Happiness of Saints in this life") 

Greatly rejoice (21)(agalliao) is the same verb Peter used in v6 (Click word study) picturing one so filled that they feel like skipping around "jumping for joy." As discussed earlier, agalliao depicts jumping and shouting for joy which cannot be contained. Remember the context - these saints are experiencing fiery trials! And yet the present tense indicates that jumping joy was their continual experience. Peter is not saying that we should rejoice over the circumstances, but that we can rejoice in the midst of them by focusing on Jesus Christ and our future with Him. Joy resides in the unseen Jesus, the Source of joy.  Future hope fuels present joy, independent of fiery trials. Stated another way, present joy based on a future hope can fuel steadfast endurance in fiery trials. Every trial we experience can help us learn something new and wonderful about our Savior. If Abraham had refused the trial in (Genesis 22), he would never have come to know God as Jehovah Jireh His Provider.

Joy (5479) (chara) is a feeling of inner gladness, delight or rejoicing.

Secular dictionaries define joy as the emotion evoked by well-being, success, or good fortune or the emotion evoked by the prospect of possessing what one desires. The world's definition of joy is therefore virtually synonymous with the definition of happiness, for both of these "emotions" are dependent on what "happens".

Certainly there is joy in human life, such as joy when one experiences a victory  (" We will sing for joy over your victory, and in the name of our God we will set up our banners. May the LORD fulfill all your petitions." Psalm 20:5 Spurgeon's comment) or reaps a bountiful harvest (see Isaiah 9:3), but more often the Bible speaks of joy in a spiritual sense. For example, Nehemiah declared to the down in the mouth (not very filled with joy) Jews that "The joy of the Lord is your strength" (Nehemiah 8:10). Similarly, David pleaded with God to “restore to me the joy of Thy salvation” (Ps 51:12 Spurgeon's Comment). It is not surprising that joy and rejoicing are found most frequently in the Psalms (about 80 references) and the Gospels (about 40 references).

C. S. Lewis got a bit closer to the Biblical meaning when he called joy an “unsatisfied desire which is itself more desirable than any other satisfaction.” That statement is a bit obtuse but Lewis then goes on to add that joy "must be sharply distinguished both from happiness and from pleasure". Ultimately Lewis' experienced joy when he discovered that Jesus was the wellspring of all joy.

Joy then is the deep-down sense of well-being that abides in the heart of the person who knows all is well between himself and the Lord. It is not an experience that comes from favorable circumstances but even occurs when those circumstances are the most painful and severe as Jesus taught His disciples declaring...

Truly, truly, I say to you, that you will weep and lament, but the world will rejoice; you will be sorrowful, but your sorrow will be turned to joy. 21 "Whenever a woman is in travail she has sorrow, because her hour has come; but when she gives birth to the child, she remembers the anguish no more, for joy that a child has been born into the world. 22 "Therefore you too now have sorrow; but I will see you again, and your heart will rejoice, and no one takes your joy away from you. (Jn 16:20-22)

Believers have the Resident Source of joy within, for as as Paul teaches

the fruit of the Spirit is love, joy, peace, patience, kindness, goodness, faithfulness (Galatians 5:22)

Emotional fluctuations cannot disturb this Source of joy. Note Paul’s statement of this confidence

For our citizenship is in heaven, from which also we eagerly wait for a Savior, the Lord Jesus Christ (see note Philippians 3:20). 

Chara in some uses can describe the person or thing that actually is the source of the joy. For example when the angel announced the birth of Jesus he referred to this occasion as a great joy (Luke 2:10). Similarly when Paul referred to his beloved saints in Philippi, whom he longed to see, he called them his joy and crown (See note Philippians 4:1).

Joy is the blessed state that is promised to the bondservants ...

His master said to him, 'Well done, good and faithful slave; you were faithful with a few things, I will put you in charge of many things, enter into the joy of your master.' (Matthew 25:21)

Rejoice in the Lord always; again I will say, rejoice! (see note Philippians 4:4;  cf note Philippians 3:1)