Through evangelism, conquest, and missionary work.
By the early fourth century, Christians formed a significant minority of the population of the Roman Empire, estimated at around ten per cent. When Constantine became the Roman Emperor, he gave Christianity state patronage and expended considerable state funds on a major program of church building and maintenance. When German tribes were conquered, Constantine required as part of the peace treaty that they convert to Christianity. Soon Christians were to be found throughout the empire. Before the end of the fourth century, Christianity was declared the state religion of the Roman Empire, and the public worship of the old gods was banned.
The next great wave of expansion came with the European conquests in the Americas, then Africa and the Pacific region. This took Christianity to all parts of the world.
Let's not forget blood, torture, war, violence of kinds the world had never seen before.
Missionaries and monks helped spread Christianity. Perhaps the most powerful force that helped spread Christianity into northern Europe was pope. Popes sent missionaries to teach people in northern kingdoms about Christianity. Some missionaries traveled great distances to spread Christianity to new lands. One of the first places popes sent missionaries was Britain. One of the first missionaries to travel to northern Europe was Saint Patrick, who took upon himself to teach people about Christianity. In the mid-400s Patrick traveled from Britain to Ireland to convert the people there.