i can honestly say, it's all about connections. I went to korea and tutored when i was a sophomore in high school. i worked as a private tutor and in an academy (i taught 3 classes - ages 5 - 10)
normally this is not allowed, but i knew a lady who used to be an english teacher so she hooked me up with a bunch of people.

I think you will have no problem. Just a warning though, if you plan on putting up fliers in apartment buildings (that may be one of the better ways to self advertise) for private tutoring, you need to make sure you have the apartment complex manager's permission.
Also when you first start off tutoring, since you have no proof of working with ESL, I think you should start off working at a smaller, local academy (not one of those fancy ones that have a chain of them). THat way you can build up your resume, not start rock bottom.

anyways, best of luck! it's awesome!! try to put up ads in wealthier neighborhoods. they have the highest pay and they are looking for native speakers that can speak english first and a little bit of korean in case their kids need explanations. However, as you probably already know, the higher your pay, the better you are expected to do.