How did the Mycenaeans gain power?

Mycenaeans didn't "overthrow" anyone. They are simply proto-greeks; that is the bronze age society which later evolved into Greeks. The Mycenaean civilization flourished between 1600 BC and ca. 1100 BC and eventually ended with the Dorian invasion of what we now call Greece. The Mycenaeans were seafarers and established colonies on Crete and several of the Greek islands. It would be very fair to call ancient Greeks their decedents.