The film is clearly ambiguous never really giving you a complete answer to the question is James real. It seems heavily sided that he isn't real but with no conclusive evidence, here is what i found to try and make the plot and the final situation a little bit simpler to understand.
Plot
Eisenberg plays Simon, a timid, isolated man who's overlooked at work, scorned by his mother, and ignored by the woman of his dreams (Wasikowska). The arrival of a new co-worker, James (also played by Eisenberg), serves to upset the balance. James is both Simon's exact physical double and his opposite - confident, charismatic and good with women. To Simon's horror, James slowly starts taking over his life.
Seeing as how the film is based on the book by Dostoyevsky i think the best reference point is to start with analysising how that book was interpreted.
The Major trend in scholarship is that many have said that Golyadkin(Simon) simply goes insane, probably with schizophrenia. This view is supported by much of the text, particularly Golyadkin's innumerable hallucinations.
It is although never confirmed in the book but Richard Ayoade could have turned this. Unlike Fight Club bost formats do not give a major hint to this, so there is no simple way to decide, both sources have given the viewer/reader their own opinion and let them make the decision, although the general consenses is that the main character goes mad and he has both personalities, i think the fact in the film the character is called "Simon James" backs this opinion up.
At the end of the novel the main character actually ends up going to a mental asylum which also strongly leads to the belief that James is not real and just a figment of Simon's imagination.