Case in point: right now, they want to persecute Paul. They first tried with the captain of the guards in Jerusalem, but he just took Paul away for his own safety. Rather than just accept that in 'quietness,' they've taken things up the chain of command. They want Felix to question Paul himself to assess his crimes.
Paul says he only got to Jerusalem 12 days ago and he wasn't stirring the pot at all, nor do they have any proof that he was. He claims to believe in the same god as his accusers and that he follows all the Mosaic laws.
As for what he was doing in the temple, well, he'd been away for many years, see, and he just wanted to clear his conscience and make some sacrifices. When the priests found him in the temple, he wasn't rabble-rousing. He was just hanging around with a group of Asian Jews who haven't brought any charges against him. He says he's already been before the Jewish court, which found him not guilty. In fact, the only objectionable thing to Jewish law that he's done is say Touching the resurrection of the dead I am called in question by you this day (v. 21)
Felix decides he can't decide anything until he gets the captain's story, so he delays trial once again. He lets Paul out on his own recognizance, and comes to him a few days later with his wife Drusilla. After speaking to Paul for awhile, he sets him free, but warns him he may be called back in future. Hoping for a bribe, he calls Paul back often over the next two years, until he's replaced by the most perfect Dukes of Hazzard name in the history of the world: Porcius Festus (v. 27), who keeps Paul in prison as a way of garnering favour with those 'quiet' Jews.