Apologies if this is an obvious one. However, I'm a bit confused. As a Jewish man living in writing in Rome, why are his works in Greek? Did he then identify as hellenistic in some way?
Other jews writing during the time, such as Philo of Alexandria, did not.
Philo did write in Greek. The OCD calls him, next to Josephus, "the most significant figure in Jewish-Greek literature," so I don't think that objection sticks.
There is a little more to the use of Greek than /u/HinkleysBane suggests when that user saysit was a matter of reaching a large audience. Greek WAS the Lingua Franca of the ancient world, but beyond that Josephus was writing history, a Greek genre (that is not to say other societies didn't keep records of their past, but that the asking of questions about the past, comparing accounts, and rendering the results in prose was/is (maybe?) identified with a Greek project). Cato, perhaps wrongly (see Aston's biography of Cato for the argument), has been judged to be, well, to put it harshly, and anti-Greek bigot for his reluctance to use Greek, and particularly his insistence on writing his history of Rome in Latin (the first Latin history on record). For Josephus to use Greek as the language for his history is for him to use the language people expected history to be written in, much like Philo using Greek was using the language people expected a platonic(ish) philosopher should use. For comparison, both Lucretius and Cicero struggled to find Latin words appropriate to describe the Greek concepts they were attempting to popularize in Rome, when Latin was to a large extent still a minor market.
On the point of market, not only was Greek wide-spread, but it was an acceptable register to Romans for intellectual pursuits. Judaism was largely tolerated (the Jewish war being an exceptional case, and ultimately not a case of religious persecution so much as suppression of political dissidents who happened to have a religion (or an interpretation of a religion) handy for political movements), but it didn't have intellectual clout so much as antiquarian clout. By using Greek Josephus got a certain level of respectability he might not have otherwise had.
Greek was the lingua Franca of the Ancient Age. Not only was it the language of the highly developed Greeks, but this language was spread throughout the entirety of the (then) known world by the conquests of Alexander the Great.
If he wrote something in Latin, it could be understood by the Romans. If he wrote something in Hebrew, it could be understood by the Jews. But if he wrote something in Greek, it could be understood by both of those groups, as well as everyone else.