This work is often done in the spring before bug season, or at the end of summer if you store your screens for the winter season. Try to pick a warm day with a bit of a breeze to facilitate drying.
The proper way to clean screens is to remove them from the window frames or doors. If you are planning on cleaning multiple screens at one time, (and it is best to considering the set up!) it is wise to number both the screen and the window with a permanent marker before starting.
Construct a simple wooden 2X4 frame on a flat smooth deck or patio big enough to accommodate your largest screen. Cover the frame with a heavy plastic or tarp and fill it with water and a few teaspoons of dish washing soap.
Lay one screen at a time into the frame, and gently brush it with a soft brush, or a synthetic broom. Brush both directions and along the screen frame before flipping the screen over and repeating the procedure on the other side.
Lift the screen out of the frame and allow the cleaning solution to drain off before rinsing the screen off with a garden hose sprayer on a gentle setting. Give the screen a quick shake and set it against a fence or wall in the sun or breeze to permit drying.
If you are storing your screens, it is best to protect them with a piece of cardboard or plywood and cover them with an old bed sheet or plastic tarp to keep them from getting dusty before the spring.
Before replacing your screen, using a bucket and sponge or clean rag, wipe the frame of the window to remove any accumulated dust.
In the event you cannot remove your screen, you can vacuum with a soft brush attachment, which will remove the bulk of accumulated dust. Using a plant mister, spray the screen with a water/dish soap solution, and using a dry clean cloth, wipe off the screen. If possible clean the exterior, then the interior.