Ok, so first there are a few things to clarify. One, it is difficult to speak about "Christianity" in the time immediately following Jesus' death. Some places to look at are Paula Fredriksen's From Jesus to Christ and Daniel Boyarin's Borderlines (the second much more controversial). The upshot from these is basically that Christianity was not a religion as we think of it until some time much later (the second century, or if you're a Boyarin person really the late third). Until then, we really have to think about overlapping and often indistinguishable types of Mediterranean monotheism, some of which are Yahwist (they believe in the revelation given to Moses, by the ancestral god of the Hebrews), some of which are not (this is the time of Mithraism, the cult of Isis, etc., and a world where Neoplatonism is about to become really important). Origen and Plotinus, the founders of biblical exegesis and Neoplatonism, respectively, attended the same philosophy lectures in early third century Alexandria. So to ask "who lead Christianity" in the first century is to put the cart before the horse, as it were. There were many people who believed in many things--some of them followed the teachings of this dude Jesus, some of them thought he was somehow divine, some of them didn't, and there were many many many combinations of these (e.g., you could be an ethnic Hebrew person who believed that Christ was somehow divine but did not think that he was the Messiah, or that you could stop keeping the other laws). A lot of the conflict between these groups comes out in the letters of Paul.

This brings us to Paul. The short answer to your question is that Paul led Christianity after Christ but, as we have seen, this is not particularly sensitive to the religious situation in the first century CE Mediterranean. But Paul spread a certain form of Christ worship throughout the Mediterranean. It does not seem to bear a ton of similarity to what Jesus the man preached; Paul almost certainly never met Jesus, and had a sometimes hostile relationship with those who had. But among the various groups who thought that this Jesus dude had been something special, Paul was the most energetic carrier of the good news and the person who, in Fredriksen's rather clever phrasing, changed the message from "the good news of Christ (subjective genitive) to the good news of Christ (objective genitive)," i.e., the message was no longer the message that Jesus had spread but just the fact that Jesus had come and had redeemed us. Paul also made the fateful decision to spread this news beyond the Jewish communities of the diaspora and bring the gentiles (from Latin gens, gentilis, "nation") into the fold. That was hugely important and allowed Christianity as we know it to come into being--in fact, the idea of being a Christian as such only arises when you have to say what you are because you are Greek and not Hebrew, otherwise you were just a Christ-following Jew, which was unremarkable. It was only when Greco-Romans began to drop out of their society that they had to answer who they were and, despite some attempts to claim the mantle of the "true Israel," were clearly NOT JEWISH.

Anyway, to cut this short--there were a lot of competing groups claiming the authority for Jesus's message. Some had better historical claims to it than others. Each had their own leaders, their own texts, etc. You can trace the conflicts between them through a close reading of the Gospels (Which differ), the letters of Paul, the conflicts in Acts, and of course the so-called gnostic or apocryphal gospels.