HERE with Paul, a prisoner of Jesus Christ, and Timothy (v.1)
Most scholars believe that this letter was written by Paul; it is the only undisputed letter of Paul to a single individual. (Although the letter is said to be from Paul and Timothy, and it is also addressed to Apphia, and Archippus, it's clear from the text that it was written by Paul to Philemon.)
Paul wrote this letter to Philemon while he was in prison. Philemon must have been a wealthy man, since he sponsored the local church. One of his slaves, Onesimus, had ran away and met Paul in prison, where he became a Christian. Paul is writing to Philemon on his behalf.
-Without thy mind would I do nothing. (v.14)
If Paul opposed slavery, this would have been a good time to say so. But no, he returned Onesimus to Philemon, since he was, in Paul's mind, his rightful owner.
See 1 Cor 7:21-24 for Paul's views on slavery.
Does God approve of slavery?
-Not now as a servant, but above a servant, a brother beloved (v.14)
Paul is suggesting here that Philemon not punish Onesimus for running away (a punishment that could be execution), but rather let him serve him forever, not as a slave, but as a fellow believer. This request by Paul is used on both sides of the slavery debate. Paul asked for a slave to be set free, but he didn't command it - or condemn slavery itself.
Greetings and Salutations

1-9
1 Paul, a prisoner of Jesus Christ, and Timothy our brother, unto Philemon our dearly beloved, and fellow labourer,
2 And to our beloved Apphia, and Archippus our fellow soldier, and to the church in thy house:
3 Grace to you, and peace, from God our Father and the Lord Jesus Christ.
4 I thank my God, making mention of thee always in my prayers,
5 Hearing of thy love and faith, which thou hast toward the Lord Jesus, and toward all saints;
6 That the communication of thy faith may become effectual by the acknowledging of every good thing which is in you in Christ Jesus.
From Paul, who is in prison, and Timothy [1]
to Philemon, Apphia, and Archippus,
and the church at your house.
7 For we have great joy and consolation in thy love, because the bowels of the saints are refreshed by thee, brother.
8 Wherefore, though I might be much bold in Christ to enjoin thee that which is convenient,
9 Yet for love's sake I rather beseech thee, being such an one as Paul the aged, and now also a prisoner of Jesus Christ.
The bowels of the saints are refreshed by you, brother.
I'm returning your slave to you

10-19
10 I beseech thee for my son Onesimus, whom I have begotten in my bonds:
I'm writing to you about Onesimus,
whom I have converted while in prison.
11 Which in time past was to thee unprofitable, but now profitable to thee and to me:
12 Whom I have sent again: thou therefore receive him, that is, mine own bowels:
I ask you to receive him again,
as though he were my own bowels.
13 Whom I would have retained with me, that in thy stead he might have ministered unto me in the bonds of the gospel:
I 'd like to keep him here with me,
14 But without thy mind would I do nothing; that thy benefit should not be as it were of necessity, but willingly.
but I wouldn't do that without your consent. [2]
15 For perhaps he therefore departed for a season, that thou shouldest receive him for ever;
Perhaps now he can return and serve you,
16 Not now as a servant, but above a servant, a brother beloved, specially to me, but how much more unto thee, both in the flesh, and in the Lord?
not as a slave but as a brother. [3]
17 If thou count me therefore a partner, receive him as myself.
If you count me as your partner, receive him as you would me.
18 If he hath wronged thee, or oweth thee ought, put that on mine account;
If he has wronged you or owes you anything, I'll make it up to you.
19 I Paul have written it with mine own hand, I will repay it: albeit I do not say to thee how thou owest unto me even thine own self besides.
I'm writing this letter with my own hand.
Refresh my bowels in the Lord

20-25
20 Yea, brother, let me have joy of thee in the Lord: refresh my bowels in the Lord.
21 Having confidence in thy obedience I wrote unto thee, knowing that thou wilt also do more than I say.
22 But withal prepare me also a lodging: for I trust that through your prayers I shall be given unto you.
23 There salute thee Epaphras, my fellow prisoner in Christ Jesus;
24 Marcus, Aristarchus, Demas, Lucas, my fellow labourers.
So refresh my bowels in the Lord.
25 The grace of our Lord Jesus Christ be with your spirit. Amen.