Christianity was able to spread so efficiently through the entirety of the Roman Empire for at least three main reasons.
First, there were the characteristics of the empire itself. It was an enormous empire to be ruled efficiently given the era being discussed. There is much speculation that it had actually peaked and was in decline. Though it had become cumbersome to rule, it still set forth with some great advances in the field of communication. Many Christian believers used these advances to their advantage.
Second, we must recognize one of its most devoted followers, apostle Paul of Tarsus. He made it his 30-year mission to preach and spread the word concerning this new religion that was based not on the ancient Hebrew religion but on the coming and resurrection of Jesus. He preached in some of the largest cities of the empire. However, it was not merely what he preached, but to whom he preached. His preachings told of loving kindness, equality, and redemption. His gospel was not only for the holy and most high, but for all people - the poor, women, and sinners. Mainly, he preached not just to Jewish people, but to non-Jews, as well.
Third, there was the conversion of the Roman general - Constantine. His converting to the religion of Christianity would prove to be a deciding factor in its triumph throughout the Roman Empire.