Hey Steven. I did the Circuit in Torres Del Paine 2 years ago in early March. Got lucky with weather, rain on three days, but could have been much worse. I went Clockwise. Most people go Counter-Clockwise. I started from Refugio Lago Pehoe after taking the boat across the lake from Guardaria Lago Pehoe. This is a spectacular way to begin your time in the park rather than starting from the entrance at Guardaria Laguna Amarga.
First night camped at Refugio Grey
Second night Campamento Los Perros
Third night Campamento Lago Paine (Coiron)
Fourth night Campamento Torres, longest day (near Torres Del Paine Lookout)
Day trip past Campamento Japones to Valle Del Silencio (highly recommended)
Fifth night back at Campamento Torres
Sixth night Campamento Italiano
Day trip up Valle Frances
Seventh night back at Campamento Italiano
Eighth night Campamento Las Carretas (see below)
To end the trip I skipped the $20 dollar boat ride across Lago Pehoe and hiked out toward Lago del Toro camping at Campamento Las Carretas. I then hiked out to catch a bus back to Puerto Natales from the visitor center at Lago del Toro.
All these camps are very sheltered with the exception of Lago Pehoe (didn't camp here) and Campamento Lago Paine which could get windy. The only other camp that may be exposed is at Refugio Dickson. Your tent doesn't need to be anything special. If you follow the above advice about camps you could even use a tarp or tarptent. Here is some gear I took.
Sierra Designs Hercules Light AST (double wall mountaineering tent, OVERKILL)
Mountainsmith Wisp 30 degree down bag (never was cold)
Osprey Aether 60 liter (old light version stripped down)
Smartwool lighweight short and longsleeve crew (sometimes layered over eachother)
Patagonia Capilene 2 Bottoms (sleeping, not used hiking)
Patagonia Stretch Jackalope Pants (fine in warmer temps)
Patagonia R2 vest (never worn while hiking)
Patagonia Micro Puff Pullover (plenty warm for camp, dayhikes, extended photo stops)
Patagonia Houdini (used a lot)
Marmot Precip pants
OR Zealot Rain Jacket (temps made me wish for pit zips)
Fleece Gloves with GoreTex overmitts
Against common advice on this site I used Goretex trail runners with full length gaiters and never had wet feet as there were no big fords to deal with.
If you want to go light, and don't mind spending the money, food such as sandwiches, chips, tuna, bread, cheese, lunch meat, chocalate, soup packets, pasta and sauce are available at the refugios along the route to supplement carried food. Refugio Lago Pehoe had the best stock. I caught a nice breakfast at Albergue El Chileno on a cold rainy morning. I also hung out at Albergue Los Cuernos for a few hours to dry out later on the same day. Great way to meet people and pass some time.
I know it's a long post but I get a little excited about the experience I had there. Feel free to PM me with any other questions. I also did 5 days in Los Glaciares on the Argentine side which shouldn't be missed if you make the trip all the way to that part of the world.