Instructions
The Penguin is the boss of the second stage along with Catwoman, meaning you will have to face the both of them at the same time. When you reach them, the two of them will be standing in front of a machine while attacking you with their various throwing weapons.
Throw your batarangs at the machine. After a few hits, the Penguin will begin dropping his penguin bombs at you. Avoid these bombs.
Spin the directional platform in the middle of the screen to redirect the penguin bombs to either side of the screen, which will blow up the platform that the Penguin is on, dropping him to your level. Now, use your batarang and punches to lower his energy and beat him.