From the New Testament gospels, we can find a portrait of a Jesus wherein he certainly considered himself to be operating within "Judaism." I mean, there are any number of phrases we could select to show this (for a particularly 'conservative' one, try "Do not think that I have come to abolish the law or the prophets; I have come not to abolish but to fulfill. For truly I tell you, until heaven and earth pass away, not one letter, not one stroke of a letter, will pass from the law until all is accomplished," in Mt 5:17-18).

Scholars debate the so-called "parting of the ways" between Judaism and Christianity; but virtually everything about this is mired in ambiguity. It's probably most useful to talk about the parting of the ways in terms of when separate groups (like some of the figureheads in Rabbinic Judaism) would consider other groups to be "illegitimate," and would attempt to exclude them from their sphere of influence and various normative social practices, etc. Certainly, in the New Testament gospels, words are placed in the mouth of Jesus that were deemed heretical--beyond the pale of "minimal"/normative Jewish belief--both by Jewish figures within the gospels, and without (and this obviously continues up to the modern day, among Jews). But, in a sense, the ideology behind this was probably thought to be a corrupt form of Judaism, rather than a totally separate ideology.

There were certainly anti-Judaic trends that developed among early Christian theologians--one of the great ironies of history. I mean, although there's an obvious sense in which this is ridiculous, you have some sayings by Paul (and others) that on the surface seem to be diametrically opposed to those positive statements made about the Law by Jesus (like the quote from Mt 5:17-18 earlier). And then you have things like 1 Thessalonians 2:14-16, which was fodder for anti-Judaism.)

Addendum: there is a phenomenally difficult crux underlying everything here (and, really, underlying many issues of religion, identity, etc.): who is the person who gets to decide who is or is not a Jew (or a Christian, or a “liberal,” or whatever)? Do we look for a "normative" body of belief and practices, find all of those who conform to these, and then exclude all others? Or are the claims of the (supposed) "outsiders," that there are in fact really insiders (just as much as any other person), enough to establish their insideness, in-and-of-themselves?

Is Islam the "perfected" Abrahamic religion, over against Judaism and Christianity? Are Mormons/LDS "Christians" just as much as anyone else is?

The ambiguities of insider/outsider identity in regard to Judaism/Christianity are probably no more tangible than at the beginning of the 9th chapter of Paul's epistle to the Romans:

For I could wish that I myself were accursed and cut off from Christ for the sake of my own people, my kindred according to the flesh. 4 They are Israelites, and to them belong the adoption, the glory, the covenants, the giving of the law, the worship, and the promises; 5 to them belong the patriarchs, and from them, according to the flesh, comes the Messiah, who is over all, God blessed forever. Amen. 6 It is not as though the word of God had failed. For not all Israelites truly belong to Israel, 7 and not all of Abraham's children are his true descendants; but "It is through Isaac that descendants shall be named for you." 8 This means that it is not the children of the flesh who are the children of God, but the children of the promise are counted as descendants.

Simple answer: yes, according to the texts of the New Testament, Jesus was a Jew acting as a continuation and a fulfillment of Jewish messianic prophecies.

My first point is that there are repeated references in the Gospels and the entire New Testament to the Hebrew Bible, where Jesus (and others, like Peter, Paul, John the Baptist, and a bunch more) would quote from Jewish scripture.

Jesus was, in the view of the New Testament, the arrived Messiah, properly filling the requirements set forth by prophecies from the Hebrew Bible. Matthew 1:22-23 (that is, the Gospel of Matthew, the very first book of the New Testament), for example, is a quotation from Isaiah 7:14. Isaiah states (depending on your translation) that the messiah would come from a young woman, or more commonly, a virgin. Matthew uses this and Mary as proof that the child here is the one whose coming was foretold in Isaiah. In the second chapter of Matthew, at 2:5-6, we come across a combined quotation from Micah and 2 Samuel stating that the messiah would hail from Bethlehem, which Matthew tells us is where Jesus was born. There are around fourteen references through the Gospel of Matthew alone to various Jewish texts that reaffirm that Jesus fits all the criteria necessary to be the Jewish Messiah, not to mention the significant number of others in the books beyond.

I would contend that the myriad references to Jewish texts were attempts to legitimize him as the Jewish messiah, linking him to Judaism in a profound way.

It's also tough to ignore the references to Jewish figures alongside Jesus in the text. This quotation from Mark, known as The Transfiguration, is a good example: "After six days Jesus took Peter, James and John with him and led them up a high mountain, where they were all alone. There he was transfigured before them. His clothes became dazzling white, whiter than anyone in the world could bleach them. And there appeared before them Elijah and Moses, who were talking with Jesus. Peter said to Jesus, “Rabbi, it is good for us to be here. Let us put up three shelters—one for you, one for Moses and one for Elijah.” (He did not know what to say, they were so frightened.) Then a cloud appeared and covered them, and a voice came from the cloud: “This is my Son, whom I love. Listen to him!” Suddenly, when they looked around, they no longer saw anyone with them except Jesus." (Mk 9:2-8)

Then, of course, there's Jesus' "cleansing of the Temple" in Jerusalem in John 2:13-24. He angrily addresses the people in the temple selling various goods and changing money.

A second quick point is that, according to the Gospels, Jesus was born a Jew. Luke 2:21 tells us that "After eight days had passed, it was time to circumcise the child; and was called Jesus...." To my knowledge, really the only ones circumcising their children at this point in that region were the Jews.

Thirdly, there isn't ever really a place where Jesus says anything about creating what I would understand as a new religion. He talks of a new covenant (as does Paul later in the New Testament when discussing the meaning of Jesus' death); he talks about being the "new Temple": "Jesus answered them, “Destroy this temple, and I will raise it again in three days.” They replied, “It has taken forty-six years to build this temple, and you are going to raise it in three days?” But the temple he had spoken of was his body"; he even talks about a "new commandment" in the Gospel of John.

In several of the accounts of his death in the Gospels (all but Luke, in fact), Jesus is referred to as the "King of the Jews," and this proclamation, according to the New Testament, is what got him killed.

The references to Jewish scripture, the details around Jesus' life and ministry as related to Jewish life, and the lack of any statements from Jesus suggesting that he intended to build a new religion all point to an intended reformation, rather than an attempt to break away and build something new.

Hope that helped! Post questions if you've got 'em! :)

Sure, Jesus was trying to reform Judaism, in that he claimed to be the fulfillment of Jewish texts--but the binary nature of the question ("reform the Jewish faith" or "trying to begin a new religion") can be a little misleading.

Your humanities professor said Jesus was a "Jewish guy trying to reform his church"--but the really interesting part of that sentence is the beginning. Jesus certainly wasn't claiming to be "[just] a Jewish guy". According to the only record we have of his teachings, he claimed to be something way more important than that--so it doesn't make much sense to call him a "reformer" if that draws comparison with guys like Martin Luther, who emphatically weren't claiming independent divine authority to speak the word of God.

He claimed that the Old Testament prophets testified of his coming, that the ruling authorities of contemporary Judaism were out of sync with their teachings--and that he was the Son of God, able to authoritatively interpret God's law regardless of tradition and precedent. That sounds like "reform" of a sort, but how radical does a reform need to be before it's something fundamentally new?

To illustrate--Muhammad also claimed to be restoring the correct interpretation of the Abrahamic tradition, and leaned heavily on the Hebrew prophets, but you would hardly call him a "Jewish reformer".

Thank you so much to everyone for the answers, it is more clear to me now. Now I realize it is a more interesting and involved concept than I had originally thought.