He offers both public and private readings and has been featured on many television shows, but is there any evidence that he can contact the dead as he says?
In a word: No.
His methods are the hallmark of Cold Reading. As listed at the end of that Wikipedia article on cold reading, he does also use all the other techniques to "validate" his performance, such as:
Granted, this isn't definitive "proof" of what he is doing, but the fact that anyone who is skilled at such techniques can achieve the same results without claiming to talk to the dead would be a strong indicator. Also, the predictability (Cold reading bingo) of the words John Edwards uses is another give away (as I once read, if he's talking to the dead, why does it sound like he's attempting to interpret a charade?).
This website "Demystifying John Edwards" also addresses his techniques, as well as alleges that he used outright fraud in the editing process:
This link is from the Committee of Skeptical Inquiry that did a relatively in-depth investigation of John Edward's claims. AGAIN you see that he is using well known techniques... And this references a well known expose done on Dateline that showed how pathetic John Edwards's attempts really are.