Is it correct to use the word "in"? We say, "I have faith in me". In the same way, can we say "I don't believe in you"?
Instead of you how about you say ghosts?
I don't believe ghosts.
This means that when ghosts tell you things, you tend to think they're lying.
There is another meaning that is conveyed when the phrase is said in an exasperated or angry tone, which is that you are surprised and angered/exasperated by the behaviour of ghosts.
I don't believe in ghosts.
This means you do not believe in the existence of ghosts.
All of these meanings are applicable to the sentence when you is used.
However, "I don't believe in you" also means "I have no faith in you". Context should provide the required clues as to which meaning is relevant.