There are many international schools in South Korea, most of which are DOD - Department of Defense - schools for dependents of personnel stationed in Korea and civilians working with the army, navy, marines and airforce. We are also aware of a couple Korean children that attend these schools and a couple American kids who were not affiliated in any way with any part of the American military.
If you're American, or if you're a teacher with dependants who wants to teach in Korea you might want to look at this angle to get your kids into an international school. This is not to say that you could necessarily teach there but rather that you might be able to get your child in which would be far cheaper than a traditional international school where the tuition is pretty steep.
International schools will only accept candidates with teaching degrees and often several years experience. If your purpose in teaching abroad it to experience culture we urge you to consider public or private schools rather than the international schools for obvious reasons.