"I don't care what you do, but please do something." is the correct version.
"I don't mind" is used to express that something will not make you angry. It is often used to ensure someone else that what he or she does is not a problem for you.
"I don't mind if you don't want to join me for the concert."
"If I lose the bet, I don't mind."
"I don't care" is a lot stronger and expresses that something is not important or irrelevant.