When my French bank failed to make a money transfer to Korea in time to secure me a dorm room, I admit I was kind of worried. How the hell am I going to find a flat in Korea? Actually, not having a room in the university dorms is one of the best thing that happened to me. Maybe I had to pay a little more but at least, I am free (Korean dorms are pretty strict...) and I have a bigger place to live!
Korea University is in Anam-dong and to tell the truth, there's nothing interesting there. And even though Yonsei is in the heart of Sinchon, you have to walk a bit to get to the interesting places. Not having a room in the dorms meant I could choose where I wanted to live. And that would be Hongdae :) I didn't mind the 40min ride to school everyday.
Also, dorm rooms can be pretty small. Like very small. University dorms are still ok, but if you go to off-campus housing places, sometimes you won't even have room to open your luggage.
Anyway, back to the interesting part, I think the best way for a foreigner to find a place is through craigslist. It might be a little more expensive since it's between foreigners, but there are plenty of ads, so plenty of places to choose from, without speaking Korean. Also, since it's targetted at foreigners, the deposit money can also be slightly lower.
One thing you have to know before opting for your own place is that deposit money are incredibly high in Korea. In France, it's one month rent but in Korea... It can be anywhere from 3000$ to 10,000+$ depending on the appartment. Also the higher the deposit money is, the lower is the rent, but the problem is to get the deposit money! Like everywhere else in the world, the rent can be higher or lower depending on the district. Some parts of Hongdae can be really expensive, but the most expensive districts are in the south of Seoul (Seocho, Gangnam, etc.)
If you have a Korean friend, you can also ask him/her to take you to a real estate agency (부동산), they are everywhere, just look for this sign. Disadvantage is that you have to pay the agency fee, but you might have better places to choose from. I have visited really nasty places through agencies though, when the deposit money was low (3 million) and the rent around 400,000 per month, and I think the real estate agency was kind of bad, they didn't even make me remove my shoes before visiting the nasty places (maybe that's why it was so dirty).
An other option is to look for a room in a hasukjib (하숙집) or boarding house, but I haven't tried that option. Apparently, it's pretty nice, but if you're in Korea during the summer, make sure they have air cond.