For they all seek after their own interests: oi pantes gar ta heauton zetousin, (3PPAI) (Php 2:4; Isa 56:11; Mal 1:10; Mt 16:24; Lk 9:57, 58, 59, 60, 61, 62, 14:26, Acts 13:13, 15:38; 1Co 10:24, 33, 13:5 2Ti 1:15; 3:2; 4:10, 16)

All (3956) (pas) means one and all. Evidently the church at Rome was living in a "me generation." In his letter to the saints in Rome Paul wrote (in about 57-58AD)...

First, I thank my God through Jesus Christ for you all, because your faith is being proclaimed throughout the whole world. (see note Romans 1:8)

At one time the Roman church was interested in others. They had a reputation for sharing their faith. Now (the letter to Philippi was written about 61AD) they could care less. They appear to have lost their passion for others.

Wuest adds that...

The word “all” is strong. It means “the whole of them, one and all, all without exception.” (Wuest, K. S. Wuest's Word Studies from the Greek New Testament: Studies in the Vocabulary of the Greek New Testament: Grand Rapids: Eerdmans)

Wiersbe comments that

"In a very real sense, all of us live either in Philippians 1:21 (note) or 2:21!"

Seek (2212) (zeteo) means to strive (continually = Present tense) after with the idea of earnestness and devoting attention and priority to, in this case selfishness rather than the Savior's cause, quite a contrast with the charge earlier to

"do nothing from selfishness or empty conceit" and "do not merely look out for your own personal interests, but also for the interests of others" (see note Philippians 2:3-4)

Wuest writes that Paul did

not mean that Paul had no genuine Christian friends in Rome, but that all shrank from visiting far distant Philippi. (Ibid)

Most people are interested in themselves, their advancement, their concerns.

And do you seek great things your yourself? Do not seek them (Jer 45:5).

Barnes writes

How many professing Christians in our cities and towns are there now who would be willing to leave their business and their comfortable homes and go on embassy like this to Philippi? How many are there who would not seek some excuse, and show that it was a characteristic that they “sought their own” rather than the things which pertained to the kingdom of Jesus Christ? (Philippians 2)

Dwight Pentecost writes that...

Paul is saying a very sad thing. He is saying that as great as the need is for someone to minister in the Philippian church, no one cares — no one! Where are all those whom Paul has led to the Lord and schooled in truth in order that they might minister? They are gone. Where are those who walked with Paul and were trained to carry on Paul’s ministry? They are not here. When Paul looked for someone to go and meet the need at Philippi, Paul says, “They all seek their own.” They are selfish, too busy to go, too occupied with their own business to care, too selfish to endure what was involved in going. And their selfishness, their preoccupation with their own things, their preoccupation with the things of this life caused them to forego the privilege to minister to the saints of God that the apostle set before them. It is this that is breaking the heart of the Apostle Paul. Here are saints to be taught. No one cares. Here are wounded hearts to be bound up. No one cares. Here are men to be reached for Christ. No one cares. Here are children to be taught and trained and guided in the things of the Lord, and no one cares. No one cares except the Apostle Paul.

That which characterized the Philippians could so well characterize us. There are burdens to be borne and shared, but no one cares. Saints of God to be taught, but no one cares. Young people and children to be trained in Sunday school, Vacation Bible School, youth ministry, home Bible clubs, but no one cares. Why? Too busy? Too preoccupied with their own business? No one cares. Recently I read one of the most disturbing articles I have read in a long time. This article predicted that in ten years vast numbers of Sunday schools across our nation would have to be closed because of lack of teachers to teach. It drew our attention to the affluence of our day that makes it possible for people to have weekend homes and engage in weekend recreation that takes them away from a commitment to the Lord’s work. It anticipated a four-day work week when men will have even longer weekends, providing for greater opportunities for recreation away from the city. That means less commitment to the things of the Lord Jesus Christ. Such carelessness is not a sign of the mind of Christ. Beware lest you become one of those about whom Paul writes when he says that no one cares. The joy of sacrifice or the curse of indifference. Which is it?  (Pentecost, J. D. The Joy of Living: A study of Philippians. Kregel Publications) (Bolding added)

not those of Christ Jesus: ou ta Iesou Christou: (Phil 1:20 21; 2Cor 1:5; 5:14, 15)

Not (ou) signifies the absolute negative. There was absolutely no desire to glorify Christ Jesus. Lloyd Ogilvie remarks that most of us are overly concerned about our own affairs and allow our time to be filled by multiple demands and responsibilities. We must ask ourselves what is truly important. For Timothy, the important thing was the cause of Christ. Ogilvie says we must ask ourselves the question,

"Is what I am doing advancing the cause of Christ or have I asked Christ to bless my causes?"

Dwight Edwards writes that...

The second reason for sending Timothy is now given. No one else qualified for the task. What disqualified them? They were primarily seeking after the things which pertained to themselves, not the things which pertained to Christ. The genitives here appear to be genitives of possession. These believers were seeking after the things which belonged to them (or so they thought). They were concerned about saving. their own lives and making this time on earth comfortable. While they were seeking after their own things, the things of Christ were passing them by. How opposite to Phil. 2:5-11. (Philippians)

Dietrich Bonhoeffer said that

BUT YOU KNOW OF HIS PROVEN WORTH: ten de dokimen autou ginoskete (2PPAI): (Acts 16:3-12; 2Co 2:9, 8:8, 22, 24)

"But you know from experience his character which has been approved after having been tested" (Wuest)

"But the credentials of Timotheus are before you: you know how he has been tested by long experience," (Lightfoot)

But (1161) (de) is used here as an adversative conjunction which places Timothy's character in stark contrast to those  mentioned in the preceding verse.

You know (1097) (ginosko) (continually = Present tense) means to know from having gained experience and thus the Philippians knew Timothy's character from personal encounter, as a man who had stood the test. Do others know by their personal interaction with you that you are a man or woman who has stood the test? .

Proven worth (1382) (dokime [word study]) (Click for in depth study of the related verb dokimazo) can describe a trial, test or ordeal (2Co 8:2). More commonly in  the NT dokime refers to that which has been tested and approved and when used of a person refers to proven character or tested value. Even as precious metals were tried by fire, so believers are often tested in the fires of suffering, adversity and persecution resulting in a purified faith. Three times they had seen Timothy (Acts 16:13; 19:22; 20:3f.) and so the Philippians knew “Timothy’s record” and he had stood the test and proven his worth. Thus they should realize immediately that no “mediocre substitute” was being sent to them.

Ray Pritchard (Making God's A-Team) adds that dokime means

to be approved by passing a test. It has the idea of demonstrating under pressure that you have the “right stuff.” How did Timothy prove himself? By sticking with Paul through thick and thin. By volunteering to tackle the hard jobs. By refusing to cut and run under fire. By doing the menial tasks, the “dirty work” so that Paul was freed up to do what he did best.

Note that this kind of “proving” doesn’t happen overnight. Too many people want “instant” spirituality and overnight maturity. God doesn’t work that way. Producing Christian character takes time and effort. Here’s a simple equation:

T + D = G

T = Time, D = Discipline and G = Growth. This formula works in every area of life, whether it be weight lifting, piano playing, Scripture memory, or learning to speak Ibo. Nothing worthwhile can be conquered in one evening. You can’t “blitz” your way to spiritual leadership. You’ve got to do what Timothy did—put yourself under a good leader and then pay the price over time.

When will we learn that God is not looking for superstars? We already have too many superstars in the Christian world – people who build their careers on hype and glitz and marketing pizzazz. God wants faithful people who have proved their worth over the long haul.

Remember … you can buy talent,
but you can’t buy faithfulness.

In his first epistle, Paul had written to the Corinthians

"so that I might put you to the test (dokime), whether you are obedient in all things." (1Cor 2:16)

In his second epistle Paul gives a commendation to

"our brother, whom we have often tested (verb form dokimazo) and found diligent in many things, but now even more diligent because of his great confidence in you." (2Cor 8:22)

In this verse Paul is referring to an unnamed brother in Christ whom he had tested and was willing to trust dealing with the collection for the saints in Jerusalem.

THAT HE SERVED WITH ME IN THE FURTHERANCE OF THE GOSPEL LIKE A CHILD SERVING HIS FATHER: hoti os patri teknon sun emoi edouleusen (3SAAI) eis to euaggelion: (Php 2:20 1Cor 4:17 1Ti 1:2, 18, 2Ti 1:2 Titus 1:4)

"that as a child to a father, with me he served as a slave would do in the furtherance of the good news" (Wuest)

Served (1398) (douleuo from doulos = bondservant) (Click for in depth study of related word doulos) means to be in a position of a bondservant and act accordingly. It describes a servant who willingly commits himself to serve a master he loves and respects. The truly surrendered doulos (which Paul certainly was, and with whom Timothy was equal souled) had no life of his own, no will of his own, no purpose of his own and no plan of his own. All was subject to his master. The bondservant's every thought, breath, and effort was subject to the will of his master. In sum, the picture of a bondservant is one who is absolutely surrendered and totally devoted to his master. What a picture of Paul and Timothy's relation to their Lord! What an example for all believers of every age to emulate!

Write it down in large letters:

The world looks for winners …
God looks for servants.

Matthew Henry adds that...

The phrase "in the gospel" means in getting the gospel out to the Roman world. Sharing your faith to a world who never heard of Christ is rough business. But he was willing to put himself on the line with all its risks and dangers. He cultivated character through trusting God in difficulty. (Notes on Philippians 2:22)

Dwight Edwards elaborates on Timothy's proven character, writing that...
 

These believers didn't have to be told of Timothy's character, for they knew of it by experience. First, he was a man of intense loyalty. He related to Paul as a son to his father, and he labored with him accordingly. Secondly, he was one of intense servanthood. We see this in "served"... as a bondslave. Thirdly, we see intense commitment. He served not in the pursuit of his own things but in "the gospel." (Philippians)

MacArthur adds that