There are two camping in Athens, one in Kiffisia and one on the road to Corinth, about 20 minutes by bus from the city center, but then frankly it's not too engaging because the area is trashy.
It is recommended only for those who came to Greece with a camper. I would advise others to take a bed in a hostel or in a youth hostel around Omonia, at least you'll be at the heart of the city.
Athens is not really the place to do camping, wait to be in the islands.
The camping of Nea Kifissia is open all year long. It is located in the north of Athens about 15 km from the center (60 Potamou and Dimitsanas). The nearest metro station is Kifissia, terminus of the green line that goes to Piraeus (15 minutes by bus).
The campground is well shaded and well maintained, it has a pool, a bar and a restaurant.
Tel: 0030 210 79 80 755
http://www.camping-neakifissia.gr/