I decided that on days where I am working out my chest, that I will end the day with pushups and put a plate on my back. Is this a good idea?

You'll need someone to hold it in place. Else, use a backpack.

thanks, I was just wandering if it were possible.

If you have a good chest workout, I don't think you will need the extra weight on your back to begin with, but if you have to then I agree that you should load some weights in a backpack (keep it higher on your back though, you don't want to put extra pressure on your lower back like most people wear them)