In this context, the verb "to come through" means "to show" or "to be communicated".
So, "Let your personality come through" means "Allow your personality to show or to be communicated".
Imagine that you are going for a job interview, and someone is giving you some advice for the interview. They say "Let your personality come through". They are telling you not to hide your personality, but to allow the interviewers to see it.
Hope this helps.