Applying some deductive reasoning may suggest that he did not exist.
First, ask yourself how one of the wisest men who was the mentor of one of the greatest philosophers and in turn a great writer, never wrote a thing? We would have to deduce that he was either illiterate, or unwilling to write, but that would be inconsistent with the personality of someone who is a teacher. Have you ever met a professor, for example, that never wrote anything? Given that his immediate protege was an accomplished writer, this seems strange. As the student often imitates his master, it makes no sense that the protege was a writer and the master was not.
The second item to look at is that even though many use the writing of his students as "evidence," there were only two students who "met him" during his lifetime. Wouldn't someone as reputable as Socrates have more admirers, more people writing about him? Or even more students? The wisest man in Athens could not have had only two students.
The third thing to consider may well be that even those who allegedly met the man have varying descriptions of him, as well as the fact that even his greatest student writes of him in sometimes contradictory ways.
In my estimation, the support for his existence is actually rather weak. But we have to then ask ourselves why would Plato or anyone else make up a man and not take credit for their own philosophical works? This may be given away in the story of Socrates, as he was allegedly hung for his beliefs. This would in turn imply that philosophers were liable to be persecuted or killed because of their beliefs, and what better way to give yourself plausible deniability, than by creating a character on which all your philosophical work is based? For example, if Plato were to be confronted by authorities, he could simply explain that the views written in his works were not his, but the thoughts of his master which he had simply recalled in his works. Consider this, that in later times as in the enlightenment period, writers would often use pen names or nom de plume in order to avoid the authorities.
While my views are skeptical in their nature, it should not take away from the ideas of Socrates, whoever they may truly belong to. But just because we rely on these ideas, does not mean that it should be taken for-granted that the man behind them existed. I would say however, that it is ironic that so many who study philosophy and would otherwise be logical and skeptical people, refuse to examine this issue and essentially believe in his existence on what amounts to little more than hearsay. No written works, no burial site, no way of knowing if he ever really existed, yet many people take it as fact and take offense if someone questions this premise. But if people can have faith in other things, maybe they can too believe in Socrates?