Although people travel to Athens all year round, the best time to go is not the summer, when temperatures get up to 40C in the shade, and the air becomes so thick with humidity and pollution that even standing still is hard work. Spring and autumn are cheaper, and popular, especially in March and October, when the weather is at its most pleasant.
In winter, Athens is still relatively warm, although there is more rain, also tourist numbers are lower, so you don't have to queue so much to get into the attractions.