1. The Jews persecuted Christians for Blasphemy
From the perspective of the High Priest, the followers of the Way were violating the primary profession of the Jewish Faith: "Hear O Israel, the Lord your God is One."
Jesus, in claiming to be God, was, according to the High Priest, blaspheming. Those who followed him would, to the Jewish mind, have been guilty of bifurcating God - of making God out to be polytheistic - of being two.
The disciples specifically were charged with teaching in the name of this "heretic" and "blasphemer." This had the effect of subverting the authority of the High Priest and causing social discord. And yes, sedition and insubordination are valid charges in a kingdom, regardless of whether or not they are in a democracy like ours today. Acts 5 records this exchange:
27 The apostles were brought in and made to appear before the Sanhedrin to be questioned by the high priest. 28 “We gave you strict orders not to teach in this name,” he said. “Yet you have filled Jerusalem with your teaching and are determined to make us guilty of this man’s blood.”
2. The Romans persecuted Christians for Atheism
For the Romans, however, the charge was different. The Romans saw the Christians as being A-Theos (literally atheists) by denying the gods they grew up with, thus inviting destruction amongst everybody by their disbelief. In rejecting the "gods" of their land, they had become 'godless Christians' and atheists. Unlike the Jews, who had special exemption from the requirement that they believe in "gods," the Christians, to the extent that they were not being Jews, were not privy to these exemptions, because the Christians had no historical basis on which to base their 'religio'.
As an example, Acts 19 shows this in practice. To wit:
23 About that time there arose a great disturbance about the Way. 24 A silversmith named Demetrius, who made silver shrines of Artemis, brought in a lot of business for the craftsmen there. 25 He called them together, along with the workers in related trades, and said: “You know, my friends, that we receive a good income from this business. 26 And you see and hear how this fellow Paul has convinced and led astray large numbers of people here in Ephesus and in practically the whole province of Asia. He says that gods made by human hands are no gods at all. 27 There is danger not only that our trade will lose its good name, but also that the temple of the great goddess Artemis will be discredited; and the goddess herself, who is worshiped throughout the province of Asia and the world, will be robbed of her divine majesty.” 28 When they heard this, they were furious and began shouting: “Great is Artemis of the Ephesians!” 29 Soon the whole city was in an uproar.
Here you see the charge - 'he is saying our gods are no gods at all!' (and, incidentally, you're not very good for business either!)
3. The Jews persecuted the Christians by using Roman charges
Since the Jews had no law to put a man to death, the next best thing was to charge that these Jewish non-Jews (and the Gentiles that accompanied them) were not privy to the Jewish exemptions and were subject to the Roman penalties against Atheism. As such, they just had to differentiate the Christians from the Jews in order to aid in their persecution.
Since the Christians held that there was no God but God the Father, incarnated through God the Son, they were subject to the invective of Rome by means of the Jews.