OK for Seoul types heading over (Jonathan that's you!) dearest Ilsanbro (Corey) has left this sage advice:
"take a bus from Express bus terminal (orange line subway) to Sokcho. Buses here leave every 30 minutes and the trip is about 3 hours costing you 22,000won."
From Sokcho you'll have to city bus it to Seoraksan National Park, but get off at Seoraksan Ya-Yeong Jang/ 설악산 야영장 (this is the camp ground) IF YOU WANT TO CAMP. Go right to the entrance (ip-gu/ 입구) if you are going to jump straight up to Pisondae shelter (40min hike from the enterance).