Bring an umbrella for any time of the year. Even in the summer it can provide you with shade. Alot of Koreans use them here. You can buy one here also. They are most useful. It's hot hot hot in the summer and cold cold in the winter w/ rain and mild weather in between.
Avoid Seoul during the wet monsoon summer roughly from June to August. It can rain non stop all day long with heavy downpours.
Not a good time for snapshots, as demonstrated by the picture from Seoul Tower.