Pack a proper winter coat for December to February, when nights fall below freezing, and light clothes plus an umbrella for July and August, the rainiest months. Spring and autumn need layers.
Seoul does real winters. From December to February nights fall below freezing, so a warm packable down jacket plus a hat and gloves is the difference between enjoying the city and hiding indoors. Summer is the opposite: hot, with July and August the rainiest months, so pack a compact umbrella.
The city is hilly and best seen on foot, so comfortable walking shoes matter. Add sunscreen for summer, packing cubes to keep layers organised, a power bank for navigation, and a plug adapter, since Korea uses European-style plugs at 220 V.
Autumn (September to November) is dry and mild, and one of the best times to visit. Seoul is hilly and best seen on foot, so bring comfortable shoes. South Korea uses Type C and F plugs at 220 V: you need an adapter.