"As Paul discoursed on righteousness, self-control and the judgment to come, Felix was afraid and said, 'That's enough for now! You may leave. When I find it convenient, I will send for you.'" NIV translation
Paul told Felix about righteousness, self-control, and the judgment. That made Felix very uncomfortable and so he sent Paul away. Paul just boldly proclaimed the Word of God and allowed God to convict Felix about his lifestyle. Do you find that many around you are uncomfortable when you talk about the things of God? That is a good thing in that as we live and speak the things of God, He will convict their hearts and they may repent and be saved.
"At the same time he was hoping that Paul would offer him a bribe, so he sent for him frequently and talked with him." NIV translation
Felix faced an internal conflict as the Word of the Lord convicted him and made him uncomfortable yet his greed made him seek a bribe from Paul. In seeking a bribe, he continued to call for Paul and hear the Word of God. Paul simply kept on faithfully sharing the Word.
God can use all things for His purposes and, here, He even uses the greed of this immoral man so that Paul could share about Jesus. It is a reminder that we need to seek the "immoral men" around us and trust God to do His work in them through us. We also notice that Paul did not compromise his faith to build a relationship with Felix before sharing the gospel. In the same way, we are not to compromise and do the same things as the "immoral men" in hopes of winning them to Christ. We are to hold fast to our faith, be different, and faithfully share the truth of God's Word.
"When two years had passed, Felix was succeeded by Porcius Festus, but because Felix wanted to grant a favor to the Jews, he left Paul in prison." NIV translation
This went on for two years and still Felix did not accept Christ but he heard the gospel. Instead of doing the right thing, he then turned Paul over to his successor and he still remained in prison. We may ask how someone could do that to a man that they knew was innocent but in fact they did even worse when Jesus was crucified (He who knew no sin became sin so that we may have life). Paul who was innocent stayed in prison and preached the gospel to his captors so that they too may have life in Christ.