Vacuum loose dust, dirt and debris from the mattress with a vacuum cleaner attachment hose. Flip the mattress over and vacuum the other side, too.
Combine 1 qt. cold water and 1 tbsp. laundry soap into a bucket. Use a whisk to mix up the soapy water to form thick suds.
Dip a sponge into the soapy water to scoop out the suds. Treat any soiled areas of the mattress with the sponge and soapy water. Rub the soiled spots firmly, using a circular motion until the spot is gone.
Fill an empty spray bottle with white vinegar. Spray down the entire mattress with the vinegar, then sprinkle a light coating of baking soda on top. Let the vinegar and baking soda dry for several hours.
Vacuum the dry powder from the mattress. Flip the mattress over and treat the other side in the same manner. The vinegar will clean and disinfect the mattress while the baking soda will eliminate any odors.