Take the lawn chair outdoors, if it is not already. Set the lawn chair on a plastic sheet or tarp.
Put on rubber gloves to prevent touching the mildew or the bleach solution. Set a plastic bucket on the plastic sheet and fill it with 1 gallon of warm water.
Add 3/4 cup of liquid chlorine bleach and 1 tbsp. of liquid laundry detergent to the warm water. Mix the solution thoroughly with a wooden stir stick.
Dampen the corner of a cloth in the solution. Test the solution on a small inconspicuous area of the lawn chair to ensure no discoloration occurs.
Dampen a soft-bristle brush in the solution. Scrub the entire lawn chair with the solution, starting at the top of the chair and working down to the chair's legs.
Keep the solution on the chair for 15 minutes. Spray the chair thoroughly with water from a garden hose.
Wipe down the chair with a clean towel. Store the chair in a brightly lit and well-ventilated area.