There are a few things to point out that are probably not going to be explained but here goes.

First of all all the movies up to this point prior to Days of Future Past are in one timeline.  There is no "other universe" in my opinion as the other person theorized.  There are a few reasons for this, one thing that was explained as the fact that Trask was black in X-Men 3... When in fact I don't believe it his first name was ever mentioned.  I read he was loosely based on Bolivar Trask.  Another is why would Xavier see Wolverine's memories of the original trilogy if Days of Future Past's dark future is not part of the original movies universe?

Anyway back to the reason that Xavier is alive and my own theory.

At the end of X-Men 3, you see a comatose patient who Xavier transferred his consciousness to.  It's unstated whether this person is related to Xavier.  What we do know is that we see Xavier in a wheelchair and having his powers.  Now there's a few things that we can take from this including information that we know in Days of Future Past.

 - Xavier's powers come from his mutant genes
 - Beast is shown as being able to manipulate DNA using the formula to
   make him look human.  Perhaps when Xavier woke up in his new body he
   didn't have his powers, but with the help of Beast he was able to
   change his DNA so that he could use his powers again.
 - The reason that he could be in the wheelchair is because that body
   was always comatose so perhaps his body is incapable of walking to
   begin with, which could be the reason he is in the wheelchair.

In fact I believe the only reason he doesn't associate with Mystique is during the original trilogy is because he felt most likely that he couldn't get Raven back and just treated her as Mystique.

As we can see evidenced from Wolverine's original memories, you can see the flashbacks of what happened in the original X-Men trilogy when Xavier looked into his mind after attempting to use Cerebro, and at the end where he says to Xavier that history would be different than he remembered starting from 1973... We can determine that all the movies were most likely all one canon universe.  This movie serves as a way to retcon the entire series after 1973, which allows them to create new stories much like a new Star Trek parallel universe was made when Nero traveled back in time and changed the events at a certain time in the Star Trek reboot.