This occurred before Paul's conversion. As a student of Gamaliel, he was probably a Kabbalist as well as a Pharisee, and very zealous even compared to his peers. Without doubt, he was a highly educated man, and would have been a respected Rabbi among the Jews. This explains how he obtained papers from the religious Council at Jerusalem to wage persecution against disciples of Yah'shua (or Yeshua - the Hebrew variants for Jesus). He thought he was serving God by destroying heretics, who were corrupting the pure religion of the fathers: Abraham, Isaac, and Jacob.
Remember that Torah law called for death to those who followed false gods, and those who incited others to do so. Because Jesus claimed to be God incarnate, he was seen by some as a blasphemer. To worship him - if you considered him just a man - was equal to worshipping a false god.
Yes, the early Church was composed of Jewish converts at first, because Messiah was sent to the Jews first. Then the gospel was extended to Gentiles. This would have enraged the Jewish leaders further, because they viewed Gentiles as unclean, and association with them rendered observant Jews unclean as well. This is why they had a separate court for worship at the Temple.
Disciples of Jesus were called "followers of the Way" before we later adopted the name "Christian".
Also Paul eventually became a Christian and spread the word through letters sent around the Mediterranean.
Jewish is a race and a religion, Jesus was Jewish by blood
The teachings of Judaism are centered on the prophecies of a Messiah who will come to redeem the people. For a person to falsely claim to be the Messiah is the highest form of blasphemy in Judaism. According to practicing Jews, Jesus was such a blasphemer. He claimed to be the literal Son of God, the promised Messiah, their Savior and Redeemer.
A complex political history had led many Jews of Jesus' day to believe that their Messiah would be a political leader who would conquer their enemies (the Romans at the time) and lead them to glory and freedom. Jesus was not a political leader when he lived in Jerusalem. He claimed instead to be a spiritual Redeemer.
Many Jews did accept Jesus as the Christ and the promised Messiah. However, the Jews who did not accept Him saw both Jesus and His followers as blasphemers. This is why Saul (who was later called Paul, after his own conversion to Christianity) originally persecuted the followers of Jesus. Saul was a devout Jew, and truly believed he was doing God's work by persecuting those he felt were defiling the religion. When he gained a testimony for himself, however, that Jesus was in fact the Messiah, he devoted the rest of his life to helping spread the gospel of Jesus Christ to both Jews and Gentiles. Paul paid the highest price for his devotion. He was martyred for his beliefs and teachings.