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!