Jesus lived, practiced and taught as a Jew, and it’s unlikely that he ever thought of himself or his movement as anything but Jewish. His minsitry, as described in the Gospels, was conducted well within the norms of Rabbinic tradition for the time (including arguing with Pharisees and Sadducees. Arguing over arcana is a staple of Jewish practice from time immemorial).
It’s possible he claimed to be the Messiah. This claim, in itself, is not particulrly remarkable or offensive within Jewish tradition. It’s perfectly permissable to make the claim. The hard part is proving it. If you fulfill the expectations (restoring the kingdom of Israel, rebuilding the Temple, bringing world peace, returning all Jews to Israel, causing all the world to worship one God), then congratulations, you proved you’re the Messiah. If you haven’t done these things, then the response is not anger, but simply that you’re deluded, maybe off your meds, maybe just an idiot, but it’s not an affront to Judaism, and failed Messiahs have come and gone before without being rejected as Jews, or necessarily causing any hard feelings.
What does irreconcilably separate the Christian movement from Judiasm is the claim (whether it was made by Jesus himself, or his followers afterward) that he was God. In Judaism, people can’t be God, and God can’t be a person. Not even the Messiah is God, and Jesus wasn’t even the Messiah.
The idea that a pesron could be God was much more normal and acceptable among Gentiles, and Christianity fairly quickly became an almot exclusively Gentile movement, while contemorary Jews (if they had any knowledge of Jesus at all), just saw him, at best, as another failed Messiah. Reliable information about the eariest movement is difficult to come by, but it’s possible that he generated some excitement and built a following before he was killed, but for the vast majority of Jews (then and now) his death was a de facto refutation of his Messiahship. In Judaism, Messiahship is defined by accomplishment, not birthright. You’re not the M<essiah until you fulfill the prophecies. If you die before you fulfill them, well themn that’s the end of it. There was (and is) no tradition or expectation in Judaism that the Messiah will die and be resurrected, but if the claim is that Jesus will fullfill the prophecies after he comes back, then great, he will then be the Messiah.
The way that Jews view Jesus as a person and a Jew is variable. Obviously, the history of anti-semitic violence and oppression by Christians against Jews has made the figure of “Christ” off-putting to a lot of Jews. Some say he taught nothing new, was unremarkable as a teacher and possibly crazy. Others do feel sense of pride and ownership of him – recognize him as a product of their people and traditions and feel that it wasn’t his fault that his name and mission were appropriated in the manner they were after his death. Some rabbis have argud that Jesus should be reclaimed and recognized as a Jew – not just a Jew but the most famous Jew in history, and the most famous ethical teacher (arguments over the originality of that teaching is somewhat immaterial in this regard, because the teachings are still essentially Jewish either way).
I think an awful lot of Jews (maybe a majority) view him basically with indifference, though. Just not a historical figure they give much thought too.
I will say that I don’t think I’ve ever encountered any Jewish feelings of hosility towrads him, though. He doesn’t get blamed for Christianity.