First of all, it is not at all clear that Christianity did undermine or help to end the Roman Empire. This was a claim made most famously by Edward Gibbon in his famous The History of the Decline and Fall of the Roman Empire but it is not generally accepted by historians today. Gibbon argued that there were five aspects of Christianity that allowed it to undermine Rome.
First, he blamed the “inflexible” and “intolerant zeal of the Christians.” He said that the Christians were unwilling to accommodate the beliefs of other Romans and were too antagonistic towards those beliefs. Because they were not willing to go along with the rest of Rome, they helped to pull Rome’s society apart.
Second, he blamed the Christians’ “doctrine of a future life.” The Christians argued that the end of the world was near and that true believers would live in paradise afterwards but that pagans would be doomed to eternal punishment. Gibbon argues that this attracted many people to Christianity out of fear that they would be punished if they did not convert.
Third, he blamed “the miraculous powers ascribed to the primitive church.” The Christians claimed that their leaders could work miracles in God’s name. Gibbon argues that many gullible Romans were converted to Christianity because they believed the Christians’ claims to be able to work these miracles.
Fourth, Gibbon blamed “the pure and austere morals of the Christians.” In essence, he is saying that the Christians’ moral values were not compatible with maintaining the strength of Rome. The early Christians were opposed to the pleasures of the world such as luxurious living and even, to some degree, sexual relations within marriage. They were focused on the next life and did not think that they should be concerned with things like business or anything else that was aimed at improving their lot in this life. This weakened Rome because too many of its people stopped doing things that would have helped the economy and the empire remain strong.
Finally, Gibbon blames Christian unity, which he says created a new, Christian state within the Roman state. Since the Christians were so dedicated and so united, they were able to spread their religion across the empire. As we have seen, Gibbon argues that Christianity was bad for Rome, so clearly its spread would also be bad for Rome.
In these ways, Gibbons argues that Christianity undermined Rome. We should note, however, that modern historians do not typically agree with Gibbons’ claims.
As far as the persecution of Christians goes, it was rather sporadic. Most of the persecution was done by regional governors and was not really systematic. Christians were generally required to do things like making offerings to the Roman pagan gods. If they refused, they were punished, often with death. However, this punishment was not necessarily very common or widespread. The worst period of persecution was that of the “Great Persecution” which lasted from 303 until 313.