This achievement is earned by completing the second episode in the hero side, "Power Crazed Penguin." It consists of five episodes, which are:
1. There She Goes Again
2. Batboat Battle
3. Under the City
4. Zoo's Company
5. Penguin's Lair
It's not terribly difficult all things considered, and doesn't require getting anything else out of the levels (power bricks, super hero status, or minikits). Just complete the five levels, simple as that.
In doing this, you'll also get the final suit in 2-3 for the "Nice Outfit!" achievement as well.
The final boss fight is a fight against Penguin and Catwoman where you have to move a panel to get the penguin bombers to blow up parts of the machine, then take out the two villans. Once they're down and out, the achievement will unlock.