Before I add my less than two cents about Christianity's historical anti-Semitism, let me start by hrsaying up front that since the holocaust the churches have done an about face. They simply (and thankfully) are no longer the havens for Jew hatred that they once more. I have been accosted in the streets for being a Jew. Online, I have been told the most unimaginably horrible (and ridiculous) things. But I have NEVER been called a Christ killer. The anti-Semites I run into online are Muslims and leftists and neo-Nazis. Not Christians.
Christianity began as a Jewish messianic sect. I don't think it had to go anti-Semitic. Why did that happen?
Jews were from the beginning an embarrassment for the Church. As the chosen people of God, we had been spoon fed God's revelation. We should have "known" the truth, we should have "recognized" the messiah. It was a big, big problem that we rejected Jesus. It hit the very credibility of the Christian message.
And then there was a confluence of a few factor right at the turn of the century. The first thing that happened was that missionaries brought Christianity to Gentiles, brought it and brought it, until it was literally overrun by Gentiles who had no special love in their hearts for Judaism.
Secondly, Paul had gone out of his way to make sure that these Gentile converts understood that they had no need to convert to Judaism and take upon themselves the 613 laws. Unfortunately the converts (to this day) misunderstood this message to mean that no one, even Jews, should obey the 613, that the Law was a curse, and that it was abrogated (despite Jesus saying the exact opposite).
Finally, Jews were hated by the Romans because of all of our various messiah sects and zealots trying to overthrow the Empire. We were persecuted by Rome. Rome kind of threw the Christians in the Jewish pot, and it was in their best interest to try to distance themselves from Judaism. They did that in their writings, beginning with the gospels.
One has only to read the gospels to see how the word "Jews" is often used as a substitute for the enemies of Jesus, who are actually only the Pharisees of bet Shammai.
This reaches a peak in the story of the crucifixion when the story tellers actually concoct a silly excuse for Pilate, that he washed his hands, and this was supposed to have absolved him of his responsibility in the death of Jesus. Pish posh. The Roman Empire is responsible for all executions under its watch.
And the stories basically blame "the Jews." In the Matthew account, the Jews in the crowd actually say to Pilate, "Let his blood be upon us and our children." THAT verse was used down through the millennia to prove that the Jewish people are cursed, that we "deserve" all the atrocities that come our way. It single-handedly is used down through the years to blame all Jews everywhere at all times for the death of Jesus, hence the charge of Christ-killer (and later, deicide).
By the second century, the church in Jerusalem had lost its authority -- the temple had been destroyed and Jews had gone into diaspora. A rift had grown between this church and the other Gentile churches, and the Gentile church had the numbers. By the time the council of Nicea rolled around in the 4th century, the Jewish bishops were excluded. At the same time in history, St. John Chrysostom was preaching the most vile words against the Jews that the world would hear from the mouth of a Christian until Martin Luther.
The rest, as they say, is history. Once you have a cursed people guilty of murdering God, you can do to them whatever you want.