Peak Season: July to August
There will be a ton of tourists and also local Koreans from outside of Seoul coming into the city, so more demand means higher prices.
Off Season: January to April
Except for the Lunar New Year holiday, a week when a lot of Koreans like to get procedures done (the week that falls somewhere between mid-January and early February), this is the best season for hotel discounts. Winter is cold and there could be snow. March and April is when it starts getting better.
Normal Season: May to June; September to December
In addition to generally pleasant weather, fewer crowds, and some hotel bargains, spring and fall are the most pleasant times to travel around Korea.