Here is the evidence for the Resurrection of Jesus that I have seen given by Christians:
1. Eyewitness testimony in the four Gospels.
2. Paul saw a walking/talking resurrected Jesus.
3. The dramatic change in character of the disciples.
4. The disciples would not have died for a lie.
5. The rapid spread of Christianity, facing intense persecution, is evidence of the truth of its supernatural claims.
Here is my brief response to each one of these points:
1. We have no proof that the four Gospels were written by eyewitnesses. They are anonymous. The traditional authorship of these books does not appear in Christian history until the end of the second century. Could a fisherman, a tax collector, and a physician have written these theologically complex texts? Yes. The probability that they did? Very, very low.
2. In Acts chapter 26 Paul specifically states that his experience on the Damascus Road was a "heavenly vision". Visions are not reality. Yes, Paul says in I Corinthians 15 that he has "seen the Christ" but he clarifies in Acts chapter 26 that it was only in a vision. Seeing someone in a vision is not evidence of a bodily resurrection.
3. Yes, Christians soon came to earnestly believe in the Resurrection and boldly spread Jesus' teachings. But there are many natural explanations for this change in behavior than that a dead man actually walked out of his grave. For instance, a group of the disciples could have seen a man in the distance who looked a lot like Jesus and then suddenly disappears behind a building, hill, etc.. "He's alive! Jesus is risen!" And the Resurrection legend begins. This scenario is much more probable than that Jesus really did rise again. So no one lied. No one fabricated the story. The disciples truly believed that Jesus was alive again. The "full" story is not written down until 65-75 AD by "Mark". Legends can easily develop within days, let alone decades as in this case.
4. We have no historical evidence that even one disciple died refusing to recant seeing a resurrected Jesus. All the martyr stories about the Eleven are based solely on tradition.
5. Yes, Christianity spread, but so did Mormonism and Islam. Christians were not systematically persecuted until after most of the original disciples would have died of old age.