Is the phrase "it is people like you" correct?
It is people like you [who cause problems].
This is an it-cleft construction where the subject is not "people", but the dummy pronoun "it".
The noun phrase "people like you" functions as predicative complement of "is". Singular PCs are also possible as in It is Sue who likes you.
The bracketed element is a non-modifying relative clause, with people like you as antecedent of "who".