This is the final boss of the game. He will fire Purple shots that go straight at you, Blue shots that will come in from the side and go across the screen, and finally Green shots. The Green shots are what you need to spin at to fire back at him and do damage.
Each time you do damage, he changes his attacking pattern. The pattern for each wave is the same though. So all you have to do is memories what he does and you should progress through to the next wave easy enough.
Doing enough damage with the green shots will defeat him and unlock this trophy.