This slang expression originated in America sometime around the 1960s. In Chapman and Kipfer's Dictionary of American Slang 'no way' is listed as a 1960s phrase and 'no way Jose' as originating in Village Voice. Unfortunately, they don't present any other details so I can't check that assertion. The first verified citation I've found is from The Washington Post, December 1979:
"I've got nothing against robots. But no way, Jose, is this guy going to win."
Why Jose? There's no reason to think that Jose was an actual person. The name was probably chosen for the rhyme with 'no way', which predates the longer phrase.