As a major city, Seoul has attractions that can be enjoyed at any time. However, the times to avoid are probably July and August. Not only is it very hot and humid, and often rainy (in fact July is officially the rainy season and floods are possible) but as this is high summer a lot of places close down while people go to the countryside for a break; so there is less to do.

Spring is popular as the city and countryside are very pretty at this time, especially in March and April when the cherry and other trees are in bloom. The weather shouldn't be bad either, though nights are often still cold. Autumn is an even better time; the humidity is less but the weather is still warm, especially in September, and the colours of the trees and plants are beautiful. Some people also like to visit in winter, since it's possible to go skiing outside the city. It is very cold though. You can learn more about the Seoul weather here.