You could start with this:
http://matrix.itasoftware.com
using the "calendar" search month by month
with your home airport (s) to ICN Seoul Incheon International, South Korea
and "length of trip" a short range like 8-10 days.
To some extent hotel rates have similar patterns to airfares. More so big international cities with lots of tourism. But you could do something similar with
www.hotwire.com or other sites to search for the hotel patterns.
And then your local library probably has some travel guide books on Korea. I know the Lonely Planet series almost always has a few pages on weather and other travel date information as to good times to visit.