By the written word, by word of mouth and the first Christian Roman Emporer Constantine forcing it on his subjects.
Christianity did not easily spread through the Roman Empire. See the related question, "How did Christianity spread throughout the Roman Empire?"
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The roman empire itself. But disciples like Paul assisted in spreading the gospel.
The Romans who remained pagans did not contribute to the spread of Christianity Neither did Julian, who was the only pagan emperor in the Later Roman Empire
The apostles and Paul traveled throughout the region teaching others.
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There was not one person who spread Christianity throughout the Roman empire. Christianity began its expansion by the efforts of the original apostles and Paul. These men gained followers who in turn gained followers and the movement snowballed.
Famine and disease spread throughout the empire
By 325ad Christianity had spread throughout the Roman Empire and to Britain. If you google "spread of Christianity map" you will find maps showing the spread.
There was no single person who spread Christianity throughout the empire. Each one of the apostles of Jesus went to different areas to preach. Paul also did a lot of writing and traveling and converting.
Because it was the Roman Empire.
More than anyone before him, Emperor Constantine spread Christianity throughout the Roman Empire. True, Christianity was already well established, especially in the Greek-speaking east, long before the fourth century, but Constantine gave the religion state patronage, offered career preferment to those who claimed to be Christians, and began the long persecution of the pagan temples. From this time, and with this impetus, Christianity spread quickly in the Roman Empire.