Your correct when you say that Jesus was a Jew, he was born, lived and died as a Jew, Christianity wasn't even thought of in his time.
For the begginings of Christianity, you have to look at Saul of Tarsus/St Paul, who opened Judaism up to outsiders/Gentiles, eg, by saying that it is now permissible to eat non Kosher foods, and not having to be Circumcised.
The split came because the Rabbinate refused to accept this teaching, and of course, not recognising Jesus as the Messiah.
I don't have time now to go more into this, but there is quite a bit more, if Clanad, Cetti or Theland are around, maybe they will expound on this.