With this post I want to inform you about a number of possibilities to study Korean for free in Seoul:
Probably the most famous course is offered by Seoul Global Center (Programs -> Korean Class). They have weekday classes for every level twice a week. A course takes up to twelve weeks and it begins in February, May or September. But registration is sometimes two months in advance. In addition they have weekend classes for beginner of Korea.
Korea Foundation Cultural Center has free Korean courses where each class meets once a week. Unfortunately, there aren't many information about the class. It seems that each study level has a duration for weeks and then class students can move to a higher class.
Personaly, I've visited a free Korean class at the Dongguk University (동국대학교). It is called "Haram" (하람 한글학교) and it is organized by students of that university. Every Saturday from 11 am to 1 pm they offer four classes for foreigners. You don't have to register for class. Although they are no professionals, classes are usually well prepared and it is a great way to make some friends. People are very friendly and after the class, most students eat together in the campus cafeteria. Once a month they organize special events, where you get in touch with elements of Korean culture.
The last class I've heard from is every Thursday and Saturday afternoon: Gal Wol Welfare Center. Here you don't have to register, too. Just go there and enjoy studying!
Some offices of local districts also offer free Korean classes. So it's worth to visit your local office and ask for classes. The courses I introduced to you don't have any restrictions and everybody is welcome. Besides that there are classes for foreign spouses, migrant workers or businessmen.