This letter from Paul to Philemon was one in which Paul encourages Philemon to be merciful to his servant, Onesimus. Paul was not suggesting that Philemon was doing anything outside the law. Onesimus had stolen from Philemon and then ran away. According to Roman law, he could be punished to any degree up to and including the loss of his life. After Onesimus ran away, Philemon came into contact with Paul and he became a believer. Paul was asking Philemon to forgive Onesimus and to go even further by granting him his freedom. Paul was calling to Philemon's great love for God and His people to help him make a right decision. At this time about 1/3 of the Roman population were slaves. The Rome that Paul ministered in was often called the "cesspool of the world."
Someone had recognized Onesimus and had taken him to Paul while he was in prison. While there, Onesimus had become a believer. Paul sent Onesimus back to Philemon. Onesimus was told of Philemon's conversion and agreed to return. Paul hoped that these two believers would come to be united as brothers in Christ. The reconciliation of that relationship would glorify Christ showing the world how Christ changes a person and repairs broken relationships, making them even better than they were before.
In his effort, on the behalf of Onesimus, Paul reminds Philemon that he led Philemon to faith in Christ.
With faith that he would be released, Paul asks Philemon to prepare a room for him, hoping he would be free soon. Paul ends with greetings from himself and others.