The Roman empire was very suspicious of the new Christian cult because the culture had a very elevated view of the Roman emperor which had stemmed from some remarkable achievements, particularly under Augustus who was regarded as far more than just an outstanding ruler but rather a 'Divas', someone regarded as having supernatural or transcendent power. There is much evidence to show that some members of the imperial family such as Julius Caesar were treated as if divine during their lifetimes. This practice was in its pomp in the 30 years leading up to the birth of Christ and became a routine aspect of Roman colonial life. There were different cults for those who were Roman citizens and those who were not. Some for example followed the cult of Julius whilst others the cult of Augustus.
The cult was particularly strong in certain areas like Corinth and Galatia (to where the new testament books of Corinthians and Galatians relate) so these were areas where the first Christians including the apostle Paul were teaching about Jesus despite great danger to themselves. Julius Caesar was a particularly important figure in this area. Conflict inevitably arose as Christians were claiming that Christ was King and Messiah in direct defiance of Caesar. There is historical documentary evidence on how they dealt with the growing number of Christians who refused to worship the image of the Roman emperor, the authorities were fearful that Christians would overturn the existing social order.
One of the most interesting aspects is that Christians were labelled Atheists for not worshiping the cult of Rome. Four centuries beforehand the Greek philosopher Socrates had been forced to commit suicide for his Atheism and for rejecting the Athenian (Greek) state religion. Socrates was not an atheist in the sense that we understand today and nor would Christians be. The early Christian received harsh treatment because of it.
Summary and references from Christian history - An Introduction by Alister E.McGrath