Get a couple of basins or bowls, two clean light-colored rags, and a clean towel. Use a whisk or sink sprayer and hot water to fill the first basin with lots of mild soap suds.
Fill the second basin with cool water. Use one rag to rub suds into the fabric, and let it sit for a few minutes. Wet and thoroughly wring out the second rag, then rub it over the suitcase to pick up dirt. Rinse it in the basin of clean water as necessary. If the rinse water gets too dirty, dump and refill it.
Blot the suitcase dry with the towel, and then leave it open to dry. You can speed up drying by placing it in front of a fan. Don't put it away until it's completely dry, or it'll get gross and moldy.
The rags should be light-colored so you can tell if you're making the dye of your suitcase bleed. If you are, stop.