What is a Jew? Many tend to view it in the narrow scope of being an ethnicity - which is more accurately a "Hebrew". Virtually all of these interbred, we no longer know who is from what tribe, etc. That is why you get Ethiopian Jews, Mizrahi Jews, Ashkenazi Jews, Sephardic Jews, etc. You have a black man and a ginger man both claiming to be the same ethnicity - and while they may have the same ancestry if you go far back enough - as we all do even further back - this is not the modern definition of ethnicity. According to modern Jewish law, your mother must be Jewish for you to be Jewish, which is also not how genes or ethnicity works, this is simply a religious law they have invented and has nothing to do with blood.
People also tend to think that being a Jew was exclusive to those descended from Jacob, but this is also not the case, as the Lord notes the Pharisee's made a conscious effort to travel and proselytise; people converted to Judaism (Matthew 23:15).
So, what is a Jew? We can speak on a few levels - the misconception of it being an ethnicity (which is basically nonsense and which we will ignore, because even if it were true then obviously you would not cease to be Jewish), being in the false Jewish religion (Orthodox Rabbinical Judaism is descended from the Pharisee sect of Judaism - you can see from their numberless, petty rules that they have only grown worse), or being a Jew inwardly, which the modern Jews are not (“For he is not a Jew who is one outwardly, nor is circumcision that which is outward in the flesh; but he is a Jew who is one inwardly; and circumcision is that of the heart, in the Spirit, not in the letter; whose praise is not from men but from God”, Romans 2:28-29), which is Orthodox Christianity rightly practised.
The Lord Jesus and St John the Baptist told the Pharisaic Jews that they were not sons of Abraham, but of the devil, because they practise the devils works and do not have Abraham's faith, and that they should not claim to have Abraham as their father because God can raise up stones to be children of Abraham. The Jews murdering the God that saved them meant that the religion that was given to them first went out to the gentiles.
The Apostles identified as Jews only insofar as they came from the Jewish religion and in those days you could really trace your lineage, etc, it was a time of transference. The Apostles never distinguished between the Jewish and gentile Christians, and "the circumcision" party is spoken of negatively. St Paul deals with lots of the issues Judaisers brought on in his epistle to the Galatians and elsewhere, we don't need to keep the Jewish dietary laws and circumcision, their man made rules of the Sabbath, etc.
So, does one cease to be Jewish when he becomes a Christian? In terms of religion, he ceases to be of the false "Judaism", and becomes a true, inward Jew; in terms of cultural things, he would drop Jewish celebrations that rejoice in the shadow instead of the caster of that shadow as if He never came; in terms of ethnicity, if there were such a thing as Jewish ethnicity, of course not. The Old Covenant is dead, the New Covenant is living.