The next interesting thing to notice is that while Philemon was the primary recipient and the matters to be dealt with seem to pertain only to him and his family, it was secondarily addressed not only to the the rest of his family (Apphia was probably his wife and Archippus was likely his son) but to the church as a whole. This approach by Paul strongly critiques the individualism often found in western churches. The way Philemon acted in his private life was the concern of the entire community, and it was their roll to keep him accountable to do that which would bring maximum glory to God.
3: Paul opens with his usual blessing of grace and peace. He hopes that his letter mediates grace and peace to its recipients, and that they would then actively live it out.
4-7: 4-6 form what is called the 'introductory thanksgiving,' which was a standard part of Paul's letters. This one in Philemon breaks down into four parts.
- Paul expresses thanks for Philemon (vs. 4)
- Paul reports that he constantly prays for him (vs. 4)
- He explains why he gives thanks for Philemon, mentioning his faith and love (vs. 5)
- He tells Philemon the content of his prayer for him (vs. 6)
4: Paul begins by saying that he is thankful for Philemon. It's a consistent thankfulness that he expresses with a high degree of regularity when he prays.
5: For most English readers, the TNIV (or NIV, NLT, NRSV) renders this verse the clearest, 'because I hear about your love for all his people and your faith in the Lord Jesus.' 
Paul thanks God for Philemon because he has heard reports of the vibrancy of his faith and the great love which he had towards his fellow Christians. It's this love for his people that Paul is going to be banking on in the rest of the letter when he makes his appeal.
6: Now Paul gives the content of his prayer for Philemon. This verse is notoriously difficult to translate. The best translation I have seen is the paraphrase that NT Wright gives, 'I am praying that the mutual participation which is proper to the Christian faith you hold may have its full effect in your realization of every good thing that God wants to accomplish in us to lead us into the fullness of Christian fellowship, that is, of Christ' (pp. 177-8). 
The foundation of this verse is the faith. Paul was praying that Philemon's strong faith which resulted in love expressed through the strong fellowship that was present in Philemon's house church would have positive results, namely that God would transform him in his thinking and living which would enable him and his community to attain full maturity in their relationships with God and each other. This prayer is significant given what Paul's going to ask Philemon to do, implicitly, later on in the letter. This letter is practice in living out the implications of our faith in community.
7: Here Paul begins his transition between the thanksgiving and the body of the letter. Paul is refreshed by hearing about the strength of Philemon's love towards his fellow Christians. To be able to host a church in his house and own slaves Philemon must have been a man of considerable means. It's noteworthy then, that he doesn't seem to have been above serving his community, and this probably doesn't refer to a single occurrence, but a lifestyle of humble service in love.
Paul describes Philemon in glowing terms, but that doesn't mean that he's done growing. We will find out in the body of the letter, the manner in which Paul intended to push this man of God to grow even more in his areas of strength.
 Literal translations like the ESV preserve the word order of vs. 5 more faithfully than the TNIV does, however, we have a case of chiasm here (see e.g., Moo pp. 387-8, O'Brien pp. 278-9), which most English readers are not adept at identifying and may result in misinterpretation.
 Paul does not have evangelism in mind in this verse (the ESV, NIV, and NRSV's phrase 'sharing of your faith' while literal, is potentially misleading).