Recipients (1-3): In this seemingly private letter to Philemon, Paul[1] includes others in the address: Apphia, Archippus[2] and the church at his house.[3] These are to be aware of the situation and be involved.[4] Biblical sanctification isn't entirely a private matter. Others are to come alongside to encourage Philemon toward faithfulness (Heb. 3:12-13). It takes a village to raise a disciple!
Thanksgiving (4-5): Paul starts with his customary thanksgiving for his main addressee. God is the One who saved Philemon and sustains him in his faith[5] and fuels his love for others. God must be thanked for this redeemed life!
Prayer (6-7): Paul prays for Philemon that his Christian partnership[6] with believers would be energized.[7] This man had a reputation for practically living out his love for God's people (1:7). Paul prays that he will now once again practically work out this fellowship in the particular situation he will describe later (1:10-18). Paul adds that Philemon has the right Christian perspective that everything in his life is for Christ, not for his own pleasures. This mindset will be critical for what he must do: forgive his slave and cancel his debt.
[1] Paul is the author and Timothy is a silent contributor (Phil. 1:1; Col. 1:1). First person plural is only in 1:1-3.
[2] Archippus was in Laodicea (Col. 4:15-17), which places Philemon there also. This Laodicean destination explains why Paul's companions who sent their greetings to the Colossians (Col. 4:10-14) also sent their greetings here (1:23-24). The recipients of this letter were distinct from the recipients of the other letter.
[3] "You" in "your house" (1:2) is singular and the antecedent is always Philemon. This house belongs to him.
[4] Paul addresses the others with the "you" plural in 1:3, 22 and 25. The rest of these pronouns are singular.
[5] "Which" in 1:5 is singular in Greek and points to only faith here. Paul completes his thought on faith (toward Jesus), then he returns to his thought on love (toward saints). NET, NRSV and NIV also take it this way.
[6] "Fellowship [koi-nō-ni-a (κοινωνία)]" means "partnership." In this vein, Paul calls himself his "partner [koi-nō-nos (κοινωνός)]" (1:17). "Of your faith" shows that this is the partnership shared among believers.
[7] "Effective [en-er-gēs (ἐνεργής)]" is, literally, "energized." Partnership/fellowship with other believers at times can become woefully dormant. Hence, Paul prays that this would be effective/energized!