Spray your plastic lawn furniture with a hose to remove any loose debris and bugs that may be hiding underneath.
Mix 3/4 cup of chlorine bleach with 1 tablespoon of liquid laundry detergent with 1 gallon of water in a bucket.
Put on rubber gloves and dip the brush and sponge in the bleach solution and start scrubbing the chair. The sponge will probably work best for most of the surfaces, but use the brush to get into any cracks and crevices or to scrub extra tough dirt stains. After scrubbing, let the solution sit on the chair for about 15 minutes.
Rinse the plastic lawn furniture with the hose. Inspect the furniture for any more dirt and repeat if the furniture is still dirty.