For the first battle (Fall to Pieces) find the box with the wrench and have Corrosive destroy it. Snag the health and wrench, and you should do the following. Go under a cover zone, use the wrench and let Corrosive lift the cover off. You'll be trashed out, but you'll get back together once out, and grab every weapon on the next mission.
For the second battle (General Corrosive), Corossive's armor makes him inpenetrable, but you can destroy boxes to find a chip. With that chip, run to the jump platform, go on it, and go into the top section. Insert the chip, and repeat three more times. When you get the 5th chip in, you should be able to control Corrosive. Approach the giant computer, and activate it. There you go, you beat Corrosive once and for all!
In the level 15 minutes, the old droid in the jail cell will talk to you, when the mils come to get you, hit them with you're arm until they start shooting at you, when they do this hide behind the old droid and the mils will destroy him instead of you and a cleaner should appear! Now use it against those Titans!
In order to beat the Zombiebot King just wear him down using rockets until he pauses (doesn't move). Toss a grenade (coring charge) into its mouth. On Normal, it takes five grenades to kill him and after the 3rd and 4th grenades there are regular zombiebots that swarm the arena.
Beat story mode to unlock this feature.
To unlock General Corrosive for multiplayer, collect all of the speed chips in the game.