The first form of persecution was done more by the Jews. The Jewish leaders were the ones who had Jesus arrested and tried, then they turned him over to the Romans to be crucified. The Jews began persecuting the Christians before the Emperors did. The first martyr for Christianity was Saint Stephen. The word martyr is derived from the Greek word "martyros" which means "witness". In the bible (Acts 6:8-7:60) the story of Stephen's martyrdom is told. Stephen is arrested by the Jews, put on trial by the Sanhedrin for blasphemy, and condemned to death by stoning.
Christiantity is a sect of Juadiasm.
The empeorors had little problem with Judiasm until Claudius came to power. When the Jews began making trouble due to the Christians, Claudius expelled the Christians from Rome.
The next emperor to come to power was Nero. Nero is most well known for his persecution of the Christians. The Roman empire was almost completely pagans. The Pagans did not like the Christians. A trait of Christianity which Christians emphasized was that it was a revolt "against the old ways against the extreme conservatism of pagan intellectuals" (Hillgarth, p.17). This provoked strong opposition by pagans. It drew a large amount of lower class. Rumors began that in the secret meetings the Christians had they would commit acts of "incest, murder, and cannibalism" (Goodenough, p.36). In 64AD a great fire swept through Rome. Nero decided to use the Christians as a scapegoat to blame for the fire. They were an easy scapegoat since the torture of unfortunate people was a popular form of entertainment and the Christians were lower
class.
Nero wasn't the only Emperor to persecute the Christians. Emperors didn't like the Christians for other reasons other than rumors and scapegoats. The Roman civil religion asked for public loyalty to the state and to the traditional pantheon of Roman gods and goddesses. By following roman religion and not praising roman gods it was seen as treason.