Did Jesus Christ exist?
I'm not going to say whether or not Jesus Christ existed, simply because I think it would an impossible determination to make. I will say however, there are some certainties on this depending on how you define him. Was there a person named Jesus Christ who was the son of God, performed divine miracles, was crucified, and rose from the dead 3 days after? Most certainly not. However, if you simply ask was there a person, represented by the figure in the Bible, by the name of Jesus, with the title of Christ? I think many atheists jump to a hasty conclusion by simply saying no.
History, as a history professor once described to me, is a recording of a series of events, one leading to another. It's cause and effect. The effect then become a cause of a following effect and so forth. The events recorded in history are accepted to a degree depending on the amount of data they coincide with which can lead up to modern day. Reliable history is tangled with the rest of it in a series of complex webs that circle the globe so that recorded history is more than just the cause and effects on a chronological landscape, but a geographical and political one as well, which drives the records of human civilization.
Sometimes however, there are events which are simply an island of data sitting in a sea of contemporaneous history. Sometimes, they may even have a few connections to the rest of history, but the less they have, the more they become an island of misconnection and the more subjective the claims turn out to be. These islands might even have far more sources than some of the data in the rest of history, but at least the latter can shown to have had direct impacts on other parts of the large web of events.
The existence of a figure known as Jesus Christ is one of these islands. The likelihood of his existence, I believe, is about as supported as the existence of Paris and Helen of Troy. There's an important distinction to make here however, which is to say that Paris and Helen were both royalty, which are generally well documented figures, and Jesus Christ was the son of a carpenter.
History does not follow the son of carpenters. In fact, the name Jesus, at the time, was about as common of a name as John is today. So if you were to say that there was never a person named Jesus who was the son of a carpenter, you may be stretching your chances here. Perhaps there was even a poor carpenter's son who caught the attention of some royal guards for doing something considered taboo for the time period. But I don't think we can seriously claim that Jesus Christ, as a normal person who's stories got exaggerated, didn't exist, because history doesn't follow the sons of carpenters and it would impossible to say.
I think it's safe to say however, that he wasn't the son of God, and he didn't heal a blind man by putting mud on his eyelids.