Actually Paul died in November of 1966 due to a fatal car accident after leaving the Abbey Road recording studios.

The evidence for this claim are clues that exist throughout the songs recorded by the Beatles.  However, a brief synopsis of the events indicates that Paul McCartney apparently left Abbey Road Studios after an argument, during which he crashed his car into a lamppost.  The cause of the crash was due to his being distracted by a pretty meter maid.  After his death, apparently MI5 (British Secret Service) was concerned that this news would result in mass suicides by Beatles fans.  Accordingly they assigned a handler to the Beatles codenamed "Maxwell" who would be responsible for ensuring that the public was unaware of the event.

The winner of a Paul McCartney look-alike contest, William Shears Campbell, was recruited to take Paul's place and subsequently had plastic surgery to complete the transformation as Paul's double.

It has been alleged that John Lennon was ultimately assassinated to prevent him from disclosing this "truth" after years of conflict with MI5 regarding the clues provided in all the songs.

Clues that were left in the songs include:

1.  The phrase "Turn me on dead man" is repeated when playing Revolution #9 backwards on the Beatles White Album.
2.  The distracting meter maid was written about in "Lovely Rita".
3.  The car crash was written about in "A Day In The Life".
4.  The MI5 handler, "Maxwell", was presented in "Maxwell's Silver Hammer".
5.  At the end of "Strawberry Fields Forever", John is heard to say "I buried Paul".
6.  The replacement double is also referred to on the Sergeant Pepper's song as "Billy Shears".
7.  Paul is wearing a black carnation while the others wear red in the tuxedo photos in the Magical Mystery Tour album.

There are numerous other references and there was also the Abbey Road album cover which provided some visual evidence.

The cover is depicted as symbolising a funeral procession, with "John, dressed in pure white symbolises the preacher or heavenly body. Ringo, dressed in full black symbolises the mourner. George, in scruffy denim jeans and shirt symbolises the gravedigger and Paul, dressed in a shabby, out-dated suit and barefoot symbolises the corpse."

So we have information about events that happened, the intervention to prevent this news from becoming public, and the evidence of clues left by the other Beatles to try and get this information out.

Except for one little problem:

None of it is true.  That's the beauty of these type of hoaxes or conspiracy theories because they are held together with a kind of internal consistency so that no matter how absurd the claims, there's an explanation.  By itself, that should be sufficient reason to be suspicious, since rarely are real world events that fully documented or sourced.  In addition, even a direct statement from Paul McCartney himself could be dismissed since, by definition, he was an imposter.

One could certainly go through and question each of the pieces of "information" to establish whether they were credible or not, but invariably there would be an explanation as to why the answer would be insufficient.  Of course, there is also the deep analysis that invariably occurs by someone cobbling together disconnected pieces of information.  In general, a perfectly documented case.

Except for another little problem:
LaBour and Gray were astonished when the story was picked up first by newspapers in Detroit, then Chicago, and by the weekend, both coasts. Beatle scholar Andru J. Reeve, opined that LaBour's story was "the single most significant factor in the breadth of the rumor's spread.

LaBour also participated in a RKO television special that featured celebrity attorney F. Lee Bailey conducting a mock trial in which he examined various expert "witnesses" on the subject of McCartney's alleged death. LaBour told Bailey during a pre-show meeting that he had made the whole thing up. Bailey responded, "Well, we have an hour of television to do. You're going to have to go along with this."
Unfortunately, for true believers, even this is not sufficient evidence to discard the story they want to believe.  Once such an idea becomes ingrained in our belief systems, then nothing short of an almost religious-like epiphany is going to change someone's mind.  

From this it isn't difficult to insert the "Moon causing earthquakes, the LHC wrecking the world, a faked moon landing or Mayans crashing the stock market" in place of "Paul McCartney is dead" and you will understand why the only response that can be given to this type of reasoning is to make fun.  Then again, perhaps we're just making fun to keep you from finding out the truth.