I recently replaced the internal parts of my toilet and ever since I did when I flush the toilet, I can hear the cold water pipe bang behind the wall downstairs. How do I fix this problem?

You could try placing a piece of styrofoam behide the pipe and wall did you use copper piping . Here in canada they allow pex which is a flexible water piping that can be attached with fittings to go over the copper to.

I am assuming that you hear the banging when the toilet shuts off. The sudden increase of pressure causes the pipe to bang against something. If you have access to the pipe (anywhere) install a bumper or shock absorber.