The word 'got' serves a different function in these two sentences:
'I got dumped' is the informal version of a straightforward passive construction. It is an alternative to 'I was dumped'. If you say 'I got dumped' or 'I was dumped', this is the same as saying 'My boyfriend dumped me', meaning that your boyfriend ended your relationship.
If you say 'You got me dumped' to someone, it means that this person did or said something that resulted in your being dumped. Let's say that a friend spread some untrue gossip about you. Your boyfriend heard this gossip, believed it, and ended your relationship. In this situation, you'd say to your friend: 'You got me dumped'. But she'd probably be your ex-friend by now!