Køge works year-round, but the town is at its best in late spring and early summer — May to June — when the weather is good, the crowds are manageable, and the market square is in full swing. The Køge Jazz Festival in August brings a lot of people from Copenhagen for a weekend and gives the town a festive atmosphere, but it also means the accommodation fills up.
Winter visits have their own appeal. The medieval streets are quiet, the museums are uncrowded, and the town feels like it belongs to itself again. The beach is obviously not the draw in January, but the historic centre is perfectly walkable in a good coat.
Avoid arriving on a Sunday if you want to see the market — it only runs on Wednesdays and Saturdays.