Claudius was emperor during the time of most of Paul’s ministry. He was peaceful, and the Gospel could spread without much hindrance. However, it was during his reign, as mentioned in Acts 18:2, that he commanded all the Jews to leave Rome. This is why Priscilla and Aquila moved to Corinth and that is how Paul met them. It seems at that time, the casual Roman could not distinguish Judaism from Christianity. Christianity was still very much Jewish, and the normal observer would only see it as nonconformity within the greater Judaism.
Most of this is from this F.F.Bruce article. After introducing Claudius and his issues with the Roman Jews, he decides to mention the Jews in Alexandria. They had moved there during the Babylonian Exile, and I suppose after Pentecost, some Christians started spreading the Gospel there. It seems like the Alexandrian Jews had higher privileges than Jews in the rest of Egypt. The Egyptians wanted to fight for their rights, but their ruler, Dio, knowing about Claudius, did not expel the Jews but put restrictions on their assemblies.
The historian, Suetonius made the interesting observation that some man named Chrestos instigated the riots in Rome. He knew that the Jews and Christians argued over Christ, and Bruce theorizes that he probably assumed that Christ actually came to Rome and started all the trouble. Tacitus new that Christ had not started this trouble, but he was no interested in defending the Jews, just chronicling their activities. Either way, the riots became too much for Claudius, so he expelled all the Jews.
People wondered how Christianity started in Rome. When Paul met Priscilla and Aquila, they were already Christians and active ministers. Bruce’s theory is that at Pentecost, there were so many visitors from around the world who heard the Christians speaking in tongues and who came to know Christ, that they went back home and started some churches. I assume this is how Alexandria became Christian, too.
How did Claudius start suspecting Christians and, by association, Jews? Acts says that Herod Agrippa had a desire to stop all Christianity, even putting James the Apostle to death. At some point, he must have met up with Claudius and told him about it. In abut 44, there was a law in most of Judaea that made it a capital offense to rob a tomb. Coincidence? Clearly, Claudius knew that the Christians fought over Jesus being alive again, and he knew the rumor that the disciples stole his body. Maybe the law has something to do with it. Bruce doesn’t think so, but it does make sense.
But this all leads to the time when Claudius died and Nero took his place as emperor. Rome had a major fire. Since the Christians were all kooky, Nero decided to blame them, and that is why Peter and Paul lost their lives and why many saints were martyred.
I kind of read Claudius’s misunderstandings and kind of laugh, but it’s also amazing how one unchecked fact, one misconception that goes by uninvestigated, can have dire consequences on people’s lives in the future. We must always make sure that we tell people the exact truth without confusion, and if we are confused, we must ask questions and debate until we understand.