Yearly weather forecast.
As visible in the chart above, Seoul gets quite cold in the winter (often reaching below zero), and quite hot in the summer (reaching into the 30s). Spring and Fall, therefore, make for the nicest times to visit Seoul.
Additionally, cherry blossom season can be a particularly beautiful time to visit. This is usually around March or April, depending on the year's weather (the exact dates for any given year can be easily found online).
Image source: Wikipedia, Korea Meteorological Administration
