Start by vacuuming off the dust, and wipe off loose dirt and cobwebs. To get into the grooves and small crevices, cut off the bristles on an old paintbrush, leaving about 1" of the bristles remaining. The bristles will still be soft, but they'll be stiff enough to help dust the furniture. An air compressor also works well for blowing out dust.
Use a wire brush with a scraper to remove old paint and rust from the solid portions of wicker furniture.
After wiping off loose dirt, wash the furniture with a sponge and liquid detergent. Keep the sponge damp rather than wet, and use as much foam as possible. For mildewed areas, add 1 cup bleach to the water. Wear latex gloves to protect hands from splinters while washing the furniture. When finished, rinse the furniture with a spray bottle filled with clean water. Avoid soaking the wicker: if you do, you'll have to wait quite a while for it to dry before you can repair it.
When the furniture is dry, check for areas that need to be repaired. Most loose wicker can be repaired with hot glue or carpenter's glue.
Now that the furniture is prepared, begin painting with spray paint or a power sprayer. It's preferable to paint indoors, where no dust or insects will spoil the finish. But paint outdoors if the ventilation isn't adequate inside. Use a good enamel paint suitable for wicker furniture (most paint labels list the surfaces the product is suitable for). Hold the spray can 8" to 10" away from the furniture, and spray back and forth for the most even coverage.
If you're spray-painting indoors, use a respirator-type mask: the fumes can be fairly strong.
After painting, don't allow anyone to sit in the chair for 48 hours. This will give the paint time to cure and prevent it from stretching.