If I transcribe something either in written or audio form is it my responsibility to check that I am not breaking the copyright rules or the person who gave me the original source?
You should make a statement on your agreement contract with your client that it is along the lines of the following
"I am unable to accept any responsibility or liability for the content of any material supplied to me by the client and transcribed, copied or edited by me on the client’s behalf including, but not limited to, copyright laws."
Make sure it appears on your website too, then there should be no reason why you should be held responsible!