Privates (private lessons) are illegal without permission from your existing employer - in fact, only kyopos (Can't F2-1 holders also teach privates?) or Koreans can teach privates. For all others it is 100% illegal. This is unfortunate but true. That said, pretty much everyone who is so inclined can get in on the action- charging anywhere from 20,000 won to 70,000 won an hour to grace natives with the sound of your silver tongue. Of course, since it is so often illegal, doing them involves all sorts of shifty manuvering about to avoid agents of the Korean immigration bureau, making you feel sort of like the James Bond of the Me decade.
Another method is to can skip the above, get permission from one's employer and tutor legally. (Pretty unlikely if you work in a hakwon but doable at universities and public schools.)
You should charge at least 40,000 Won unless you owe the students a favour. All you need do is not blab about it at work, and even then it's highly unlikely to rebound on you unless the boss takes it into their heads that you are giving less than all your efforts to their business as a result.