Romeo Ur-Ribitt is the boss of Hogmarsh Hollow--the third adventure.
Clear out the minions every time they spawn and keep your distance from the boss--using the default gun on the boss is effective. I wouldn't suggest melee weapons on the boss because touching the boss damages you and uses heavy attacks when you're near.
This boss level is really cluttered too, so be proactive in staying ahead of enemies so you don't get trapped and lose your health super fast. Trying to jump over all the clutter isn't advisable either, you often get stuck trying to outrun enemies, so either clear up the area or just constantly circle the room as you kite the boss.
The achievement pops after defeating the boss.
A suggestion from Dwaggienite: I tried beating him three times, and it didn't pop. When I beat him those three times, there were still minions alive. That might have something to do with it. I'd recommend making sure that all minions of the room are dead before delivering the final blow.