If you are Asian, you have a very little chance of getting a job simply because you don't look foreign. When parents pay for their kids to study under a foreign teacher, they expect to be able to identify the foreign teacher by sight.
If you are black, I strongly advise AGAINST coming to Korea. If you do manage to find a job, you will be subjected to some of the worst prejudice you've ever come across in your life. Koreans are very afraid of black people. They get most of their education about westerners from Hollywood and Spike Lee films have done nothing to improve Koreans' perceptions of black people. Most Hollywood films show black people as very agressive and violent people. When a Korean wants to insult someone, they say something along the lines of, "Your face is so black...you must be from Africa." If you are a black male, go to Japan. Japanese women have a thing for black guys. You'll probably still encounter a lot of prejudice, but you'll get plenty of dates and girlfriends.
If you look Native American, Indian, or anything other than American white bread, you will have a difficult time finding a job. If you did manage to get a job, even your young students are likely to make fun of your looks and insult you.
If you are a very strong (emotionally, intellectually, physically, spiritually) person, then I say come to Korea no matter what color you are. Koreans will never change their attitude and beliefs until they are exposed to decent people of all colors and races. It will never happen in our lifetime, but change has to start someplace. Just be prepared for the worst, and maybe you'll be one to contribute to a much needed change.