If you want to work at a public school or university, certain times of the year are better than others for spotting job announcements and, therefore, applying for jobs. Such Korean institutions begin their school year in March, and may begin advertising as early as November/December, though, to reiterate, the positions would not begin until March. Then there are the mid-year positions which usually start in late August or early September (and are therefore usually advertised during the early summer).
Private academies, however, called "hagwons", advertise and hire year round, and there are many of them. You can find the current crop of job announcements on Dave's ESL Cafe.
You will need to plan in advance to get a criminal background check, university transcripts/diploma, etc (which may take from a few weeks to several months). The university data is usually needed to apply for a job, and the background check to secure one. Make contact with your area Korean consulate (internet page?) to ascertain the details.
You need to think about what age level--from kindergarten through to adult--you are most interested in and/or qualified to teach. Public schools will usually give you one age/grade grouping, while private schools will expect you to teach multiple age levels.
I am sure others can add more to this basic information . . .
