The slaves originally believed in pagan religions, and did not necessarily worship the God of the monotheistic Christian tradition. Perhaps the Christian religion would have spread to them even if they were not slaves, since Christianity is a predominant religion in the United States. However, the blacks' unfortunate role as slaves had a direct correlation to their assimilation into Christianity.
By 1780, the Methodists had decided that slavery was "contrary to the laws of God, man and nature, and hurtful to society, contrary to the dictates of conscience and pure religion." Clearly, they were one of the early groups to realize that slavery was wrong. In 1784, they announced that they would excommunicate any Methodists who chose to continue with the practice of slavery. In 1820, the Methodists began a mission to the slaves, and targeted their religion to them. Of course, since the Methodists were kind to the slaves, they were drawn to the religion. Many slaves were also drawn to the religion because it focused on emotions, which the slaves clearly needed to express in some sort of safe haven.
For a long time, the whites were afraid to teach the African Americans about Christianity for fear that they would act out the practices of books of the Bible such as Exodus. However, when rebellions became more intense, such as that of Nat Turner, the whites became afraid. They thought that perhaps teaching Christianity to the blacks would prevent such revolts. They wanted to regulate the blacks as much as possible, so they began to open their churches to them. They felt that if they educated the blacks about Christianity, the blacks would learn to "turn the other cheek," as Jesus Christ advocated, and to accept that their position as slaves was what they were supposed to be and they should not revolt.
Some people really did care about the slaves, and they simply wanted to allow them to become part of their culture. On the other side of the coin though were some other "good" doers. They felt that the African Americans practiced pagan religions, and they wanted to convert them in order to save their souls. This practice is objectionably good, because who is to say that the African Americans did not have the freedom to practice their own religion? Chances are that the whites did not even bother to get to know the religion of the Africans, and simply assumed that they were heathens.
A famous black religious speaker, John Jasper, helped to bring Christianity to the slaves. He proposed that God was able to intervene and turn the "natural" order upside down. He used this argument because many of the slave-owning whites called slavery the "natural" order of things. Therefore, he was inspiring hope that God could intervene and save them from the terrors of slavery. Jasper was so popular, in fact, that many people would delay funeral ceremonies until Jasper could arrive to deliver a speech.
We can see that the answer to how Christianity spread to the slaves is quite complicated. There were many components to the journey. Either way, when we look back, we see that Christianity and the slaves were intrinsically linked. As aforementioned, perhaps these changes would have eventually occured whether or not slavery existed. However, that is not how history happened. History is not based on "what-if's" but instead on what actually took place.