This study was done using the Amplified Bible, which I don't have permission to publish, so I will replace the verses here with the World English Bible (a modernized version of the ASV). Because of that some of the comments may not totally make sense because the AMP has expanded explanations. I would highly recommend going to a site like BibleGateway.com where you can read the AMP version for free.
19 I, Paul, write this with my own hand: I will repay it (not to mention to you that you owe to me even your own self besides).
A.
Paul here takes yet again the same route at Jesus, taking the
debt of another with nothing to gain but friendship with a former
slave. He is bartering for the freedom of his friend in a more
practical way now, and in a more self sacrificial form. Not
just asking a favor, but offering to pay the price for it.
20 Yes, brother, let me have joy from you in the Lord. Refresh my heart in the Lord.
A.
Paul begins to appeal to the man who owes him his very being
for joy and refreshing. He found these things in the work of
the Gospel rather than worldly sources.
21 Having confidence in your obedience, I write to you, knowing that you will do even beyond what I say.
A.
Paul is confident because of his knowledge of Philemon.
Knowing that it wouldn't be a guilt trip for him to ask such a
favor, but that Philemon would most likely see it as an honor to
bless the apostle who had sacrificed so much for the Gospel. He
goes beyond simple obedience to suggest that Philemon would do even
more than what he asked. What exactly he means by this remains
a mystery.
22 Also, prepare a guest room for me, for I hope that through your prayers I will be restored to you.
23 Epaphras, my fellow prisoner in Christ Jesus, greets you,
A.
We know little about Epaphras aside from the mention of him in
Colossians 4:12 in which Paul says he was a Colossians as well.
Aside from that we know that he was also imprisoned for the
Gospel, possibly at the same time as Paul though we do not know for
sure.
24 as do Mark, Aristarchus, Demas, and Luke, my fellow workers.
A.
These men were all known by Philemon, as this was not written
to a church body. It is also possible that Paul's mention of
them to Philemon was a suggestion that they agreed with Paul's
request for the freedom of Onesimus. B. Aristarchus is most likely the same man recorded to be with Paul in Acts 19:29 and other places.
C. The only other mention of Demas is in 2 Timothy 4:10, in which Paul says he abandoned because of his love for the world and went to Thessalonica.
D. Luke is most likely the same Luke of the Gospel of Luke and Acts. As he often traveled with Paul we can reason this to be the probable case.
25 The grace of our Lord Jesus Christ be with your spirit. Amen.
A.
According to Adam Clark, the word "your" in the verse
is plural, meaning the blessing to go to all who were in the church
at his house.