What happened to Charles Xavier after X-Men 3?

Firstly, the post-credits scene from X-Men 3 hints at his survival. If for any reason you can't view the video, this is what happens (from the wiki):

After the credits there is a scene at the hospital, where the comatose man from the video that Xavier showed his students is lying in a hospital bed. Dr. Moira MacTaggart enters the room to see the sleeping patient, only to be greeted by the Professor's voice coming from him. Dr. Moira looks towards her patient in disbelief and says "Charles?".

In X-Men: Wolverine, the post-credits scene shows Xavier's return, years after his supposed death. Again, if the video doesn't play, this is what happens (from a different wiki):

Xavier returns in his original visage and his wheelchair. Along with Magneto, he solicits Logan's help against dark forces seeking to destroy mutant-kind. When Logan questions Xavier about how he's returned, Xavier reminds him "you're not the only one with gifts."

This page quite effectively explains how this is all possible. Unfortunately, the video explaining it is currently down, but to take from the synopsis:

...the Brain Dead Patient [from X-Men 3] is indeed P. Xavier, Charles Xavier's brain dead identical twin brother. When he was born his telepathy was so massive that he essentially stole his brothers brain waves. But when he was killed off by The Dark Phoenix he transferred the consciousness of his mind into his brain dead identical brother.

So, he basically took over the body of his rather conveniently handy identical brother and was allowed to continue.

It is not shown in any movie explicitly how he is alive. However, as per post credit scenes of movie X-Men Last stand ,he transferred his mind to his evil twin. You can read this another related question

Also in the end-credit scenes of The Wolverine, he and Magneto confront Wolverine at the airport where on being asked, "How's this possible?", Prof X. says, "You're not the only one with gifts." suggesting that somehow he was able to revive himself.

Also this paragraph from Wikia supports this theory :-

In an after-credits scene, Xavier spoke to Moira MacTaggert through the body of a comatose man, implying that his consciousness survived by transferring itself into the body, an action that Xavier had discussed as part of an ethics class, the question being if it would be ethically right for a mutant like himself to attempt such a transference.

One of his capability is mentioned as :-

Mind Transfer: When his body was destroyed in X-Men: The Last Stand, he transferred his consciousness into a host body; his twin brother P. Xavier.

If you want to know how he survived , then he just transferred his consciousness to his identical brain dead twin brother.

From marvel-movies.wikia.com

In an after-credits scene, Xavier speaks to Moira McTaggert through the body of a comatose man, implying that his consciousness survived by transferring itself into the body (An action that Xavier had discussed at the beginning of the film as part of an ethics class, the question being if it would be ethically right for a mutant like himself to attempt such a transference). In the directors commentary for the DVD, it is revealed that the man that Charles transferred his consciousness to was really his identical twin brother, whose mind had been destroyed at birth when Charles power's manifested.

But how he got back to wheelchair with this new body of his twin brother hasn't been explained yet.