There are two gates ahead which prevent most of the Simpsons from getting past, so you'll have to work your way across the clouds to a switch that Bart can flip. With Shakespeare down, the gates leading into Heaven will open, and the remainder of the Simpsons family will join you - nice!
Shakespeare will then appear on a different cloud - target him like before and repeat this until he's down for good. Run up and punch him while you have the chance.
He'll fall to the ground after a few shots, stunned. The best way to tackle Shakespeare is to use fire Bart's slingshot perpetually while strafing around Shakespeare to avoid his return fire.
