There aren't a lot of options for people of the Bible starting with "F." The two I found are both Roman governors who show up in Acts. To understand how they fit in, we have to start with the Apostle Paul.
Paul was a Pharasee and persecuter of the Christian faith who was converted to Christianity after Jesus appeared to him in a brilliant vision on the road to Damascus. After his conversion, he went on many missionary trips around the Mediterranian Sea proclaiming the Gospel. The account of this can be found in the book of Acts. In Acts 21, Paul has returned to Jerusalem and is arrested on trumped up charges claiming that he is causing disruption. Paul is arrested in flogged and the Roman commander is horrified when he finds out that Paul is a Roman citizen (you can't flog or crucify a Roman citizen and they have the right to a trial). In Acts 22, Paul goes before the Sanhedrin and he defends himself to them. But, there's a plot to kill Paul so the Roman centurion sends Paul to Felix for safekeeping and a fair trial.
I looked Felix up in our Bible dictionary and I found out some interesting things. First, he had three wives. Second, apparently, he felt he could commit any crime because he had friends all over the court system. Not a nice guy.
In Acts 24, Paul gets a trial before Felix and Felix keeps him in custody (minimum security though) because he wants Paul to bribe him to set him free. I told you he wasn't really a nice guy.
After 2 years, Felix leaves Paul in prison as a favor to the Jews and is suceeded by Festus.
My dictionary doesn't say that much about Festus so it sounds like he wasn't so colorful. The chief priests try to convince him to transfer Paul to Jerusalem for a trial. What they really want to do is assassinate him en route. Paul appeals to Caesar instead (his right as a Roman citizen). Festus consults with Herod Agrippa who hears the trial and decides that Paul could have gone free if he hadn't appealed to Caesar.
Thus ends our experience with Festus and Felix