Yes, "to emphasize" is the correct verb form, "emphasis" is the noun form. We just say "to emphasize X", no preposition is needed. Thus, we could say "You don't need to emphasize the H." "You don't need to put any emphasis on the H" is also correct, and here you do need "on." In ordinary speech we would just say "the H." Being more formal or precise, we would say "the H sound." We wouldn't say "the H letter." "You don't need to emphasize the H" means that you do hear the H sound, but it is no stronger than usual. If you do not hear any H sound at all, we would say "the H is silent here."