"Since the Jews constantly made disturbances at the instigation of Chrestus, [Claudius] expelled them from Rome."
Interestingly, Acts 18:2 in the New Testament also gives an account of this event:
"There [Paul] met a Jew named Aquila, a native of Pontus, who had recently come from Italy with his wife Priscilla, because Claudius had ordered all the Jews to leave Rome."
It seems clear that the Jews were in fact expelled from Rome by Claudius on account of one Suetonius called Chrestus, but who is Chrestus? Many have pointed out that Chrestus is an alternative spelling of Christus, or Christ. So it seems that Jews were causing disturbances in Rome on account of a Christ-figure (Christ being the Greek version of the Hebrew Mashiach or Messiah). Many have suggested that this refers to Jewish mobs coming against those who preached Christ Jesus. This is not an unreasonable interpretation of Suetonius' words. After all, there clearly was outrage from Jewish mobs over Paul's preaching of Jesus which are recorded all throughout the book of Acts. In the chapter previous to the one quoted above, there is an account of a Jewish mob organizing in Thessalonica in response to Paul's preaching of Christ, and even the chapter the quotation is taken from gives an account of a Jewish mob bringing Paul before the authorities because they are offended at his preaching. Based on just these bare facts, it seems very reasonable to assume that a number of Roman Jews as well were being brought to violence over the preaching of Jesus, and that Claudius might have expelled them from Rome because of it.
One difficulty with this is that Acts 18 never explains why the Jews, along with Christian Jews like Priscilla and Aquila, were expelled. Based on the purpose of Luke in writing the book of Acts (to give an account of the earliest preaching of the Gospel after Jesus' ascension, and of the difficulties the disciples encountered), it would seem that he would be interested in sharing this information. This provides one piece of evidence that Claudius was not concerned about violence over the preaching of the Gospel, but it certainly doesn't destroy the case for the position that he was. Luke could have had any number of reasons for not feeling that it was necessary to give details here.
There is other evidence that Claudius was concerned with the "Christian problem" and particularly how it created difficulties among the non-Christian Jewish population of the empire. The Nazareth Inscription is among this evidence. While there are numerous opinions on how the inscription should be dated and understood, Clyde E. Billington PhD lays out a very convincing case that it was written by Claudius in response to the Jewish claim that the Christians had removed Jesus' body from the tomb, which is why they believed it couldn't be found there. Billington's translation of the Inscription reads, in part:
"if anyone legally charges that another person has destroyed, or has in any manner extracted those who have been buried, or has moved with wicked intent those who have been buried to other places, committing a crime against them, or has moved sepulcher-sealing stones, against such a person I order that a judicial tribunal be created..."
Billington also speculates as to what the impact this inscription might have had on Christians. He suggested that Herod Agrippa I, a lifetime friend of Claudius who provided priceless information to him about how to govern the Jews in a way which would be least likely to incite violence and respect their traditions, promptly responded to this edict with what we read in Acts 12:1-3:
"It was about this time that King Herod arrested some who belonged to the church, intending to persecute them. He had James, the brother of John, put to death with the sword. When he saw that this pleased the Jews, he proceeded to seize Peter also. This happened during the Feast of Unleavened Bread."
If Billington's understanding of this inscription, posted in Nazareth where Jesus lived (and after which His disciples were called Nazarenes), is correct, then it becomes clear that Claudius was concerned with the issue of Rabbinic Jewish and Christian Jewish relations, and that he very reasonably could have expelled all of the Jews (whether Rabbinic or Christian) from Rome because of the conflicts that were erupting in response to the Gospel being preached there.