"Together" is an adverb that modifies the verb "have". In this sentence "have" means the same as "study", in the sense of taking a class. "With" is a dangling preposition. The sentence becomes clearer if you place the adverb closer to the verb it modifies, and if you give the preposition an object: "She is someone together with whom I study Chemistry, Math, and English."

It means they both attend the same classes for chemistry, math and English.