Felix was appointed by Claudius and reconfirmed in his post by Nero. He is
best known from Acts of the Apostles 24, in which he presides over the
trial of the apostle Paul. He decides to keep Paul in prison for a short
time, but postpones Paul's release because he didn't want to create an
uprising with his subjects. The Bible reports that he was deeply
affected by his many conversations with Paul about Jesus, because he was
engaged in an adulterous marriage with the beautiful Jewess Drusilla. He
had seen Drusilla from afar and been attracted to her, so he sent a
friend of his, Atomus, to pretend to be a magician and convince her to
break it off with her husband and marry Felix. It worked. She "transgress[ed]
the ancestral laws" and married him (Antiquities 20.7.2).
Suetonius also deems it important enough to note that Felix had a soft
spot for women. He writes, "Felix married three queens" (Claudius
Aside from his personal life, Felix had his hands full just keeping the peace. "Not a day passed, however, but that Felix captured and put to death many of these impostors and brigands" (Antiquities 20.7.5). "[T]he bandits whom he crucified, and the local inhabitants in league with them whom he caught and punished, were too many to count" (War 2.13.2) For example, he tricked one particular leader of brigands named Eleazar to come to him. When Eleazar, who had apparently ravaged the country for twenty years, met Felix, Felix captured him and sent him to Rome (Ant. 20.7.5, War 2.13.2). Many of these brigands and false-religionists were really anti-Roman activists, who would attack the houses of the rich who submitted to Roman rule (War 2.13.6).
A thorn in the side of his administration was the high priest, Jonathan, who was constantly criticizing Felix's administration. Felix got his highest adviser, a guy named Doras, to bring a band of brigands named the sicarii to kill him, which they did. They did it daylight in a crowded square. The sicarii began to do this all the time, however, and thus the citizens never felt safe anymore. This state of civil unrest is what Josephus believes led to the downfall of the Temple. He felt God was unhappy with the state of affairs and was punishing the people for it (Ant. 20.7.5).
Also during Felix's reign, a false prophets were roaming throughout Judaea (War 2.13.4). "[T]hey said that they would show [their followers] unmistakable marvels and signs that would be wrought in harmy with God's design" (Ant. 20.7.6). Felix punished the prophets and the followers. One particularly successful false prophet from Egypt persuaded 30,000 people (only 4,000 according to Acts 21:38) to follow him out to a mountain top where he would commant Jerusalem's wall to fall. Felix quashed this "revolt," killing 400 and taking 200 prisoner (Ant. 20.7.6).
Finally there was a major altercation between Jews and Syrians in the capital of the province, Caesarea. The two groups had long been in conflict. Felix sided with the Syrians. He killed and captured many Jews, until some level-headed Jews went to him and asked him to stop (Ant. 20.7.7). Since the quarrel continued between the two groups, he sent representatives of both groups to Nero to work it out verbally (War 2.13.7). The Jews of Caesarea were still upset at his siding with the Syrians, so after he was taken out of office they sent a contingent to Rome to accues Felix of crimes against them. Josephus feels Nero would have punished Felix, except for the fact tha Felix's step-brother Britannicus (also called "Pallas"), who was very well respected in Rome, intervened on his brother's behalf (Ant. 2.13.9).
There is one minor descrepancy regarding the reigns of Felix and Cumanus. Tacitus reports that they were reigning at the same time, but over different geographical areas: Felix over Samaria and Cumanus over Galilee. My personal feeling is that Felix had Samaria while Cumanus had Galilee and Judaea itself, and the territory was consolidated after Cumanus left office. Felix was then promoted to a greater position than Cumanus had had, which is why Josephus specifically notes that he was sent to be governor of "Judaea, Samaria, Galilee, and Peraea" (War 2.12.8).
Was Felix's administration really as plagued as Josephus makes it out to be? Tacitus doesn't seem to think so. He writes that all the civil unrest was caused by Felix, not the other way around. "Felix stimulated outbreaks by injudicious disciplinary measures," he writes (Annals 12.54).
Felix's connections in Roman government helped him greatly. While at a trial before Legate of Syria Ummidius Quadratus, Cumanus, whom Tacitus says was as culpable as Felix in poor administration of Palestine, "was condemned for the irregularities of both [Felix and himself]," Felix was actually allowed to be a judge in the trial as if he had done nothing wrong (Annals 12.54). As Suetonius noted, Felix was one of Emperor Claudius' "favorite freedmen" (Claudius 28) and it showed.
Ancient sources: Acts 23:24-26, 24, 25:14; Antiquities 20.7.1-2, 20.8.5-7, 20.8.9; War 2.12.8, 2.13.2, 2.13.4-5, 2.13.7; Histories 5.9; Annals 12.54; Suetonius, Claudius 28.