Neither. Jesus strongly criticized the spiritual ancestors of modern Jews.
So, first, "Christian" as a word refers to the people who are the followers of Christ; it originally derives from a Greek word that meant "little Christs". As a result, I don't think it would be accurate to describe Jesus as a Christian, since he wasn't one of his own followers - he was the leader that his followers were following to begin with.
Secondly, Jesus wasn't a Jew, as we currently use that word. "Jew" refers to a group of people who are spiritually descended from the doctrines of the Pharisees, who Jesus strongly criticized, and ethnically, either descended from those same Pharisees, or from gentile tribes like the Khazars that converted to their religion. We can see evidence of what Jesus thought of them in passages like the following:
I know that you are Abraham’s descendants. Yet you are looking for a way to kill me, because you have no room for my word. I am telling you what I have seen in the Father’s presence, and you are doing what you have heard from your father.”
“Abraham is our father,” they answered.
“If you were Abraham’s children,” said Jesus, “then you would do what Abraham did. As it is, you are looking for a way to kill me, a man who has told you the truth that I heard from God. Abraham did not do such things. You are doing the works of your own father.”
“We are not illegitimate children,” they protested. “The only Father we have is God himself.”
Jesus said to them, “If God were your Father, you would love me, for I have come here from God. I have not come on my own; God sent me. Why is my language not clear to you? Because you are unable to hear what I say. You belong to your father, the devil, and you want to carry out your father’s desires. He was a murderer from the beginning, not holding to the truth, for there is no truth in him. When he lies, he speaks his native language, for he is a liar and the father of lies. Yet because I tell the truth, you do not believe me! Can any of you prove me guilty of sin? If I am telling the truth, why don’t you believe me? Whoever belongs to God hears what God says. The reason you do not hear is that you do not belong to God.”
Ethnically, Jesus was a Judean - a citizen of the country of Judea. Religiously, he was a worshiper of God, though he doesn't seem to have been a member of any of the existing major sects such as the Pharisees or the Sadducees (who ran the Temple), though he was associated with the movement started by John the Baptist. Instead, he went on to found his own sect - the early Christians.