Seoul is a huge city, the 6th biggest in the world, and so therefore has so much to see. there is no way you can cover everything in the time that you have so I would suggest baseing yourself in Seoul and perhaps doing a day trip or 2 out of Seoul. Seoul is basically a lot of different towns all linked together to form 1 huge city. For example I was in Seoul last weekend and we managed to visit:
Itaewon, which is the foreigners district where there are loads of bars and restaurants from all over the world.
Myongdong, which is the food capitol of Seoul.
Insadong which is a great area for shopping and hanging out as it is very cosmopolitan and one of the main roads is closed to traffic at the weekends for a street market.
Gwangwamun to see the amazing royal palaces.
Hongdae, which is the Universtiy area and so has a great nightlife scene...so many bars and clubs.
Dongdaemun, which has a huge market and you can stroll along the man made canal that flows through Seoul.
We did all this in 3 days yet there is still so much more to see that we will visit on our next weekend there.
Seoul is up there with the great cities like Bangkok or Hong Kong in my opinion and we cant wait to go back (I live in Korea, about 4 hours outside of Seoul).
Have a great time!