Check out this site for more info about camping in Bukhansan:
Also, since this park is IN Seoul, it gets busy! I recommend you go during the week and avoid going on Saturday and especially Sunday! That goes for most national parks in Korea in the summer and fall.
Also, camping is not the same as back home. Usually there are designated areas near the parking lot of the park. The camp sites and very close to each other and you can usually park your car in very close by, if not on your site.
The national parks do not allow camping 'just anywhere'. There is a fine for this, though I don't know how much.
I have never just hiked into a park, found a nice spot by a cool stream and pitched a tent. It just doesn't happen here. Maybe other folks have had different experiences...