Drag the chairs to a part of your yard, such as a sidewalk or concrete patio, so that the water solution won't damage your lawn.
Pick an area that has more soil than grass (depending on the time of the year), if you have no access to a large concrete area. Situate the bucket chairs onto a tarp so that any mud that splashes up from the hose will not dirty the chairs.
Fill the bucket with 2 gallons of hot water, 1 cup bleach and 2 tbsp. detergent. It is important to use laundry detergent and not some other cleaning agent in case there is an adverse reaction to the bleach. Set the brush into the mix.
Test an inconspicuous part of the bucket chair by dipping the cloth into the bleach solution and testing to see if there is any discoloration from the bleach onto the furniture. Allow the test area to dry completely. Wait 15 minutes, and then inspect to see if there is any damage. If not, then continue the steps. If there is, consult the manufacturer for specific cleaning agents that would work more efficiently, or simply leave the bleach out of the solution and continue with the steps for cleaning.
Spray the chairs down with the garden hose, making sure that you saturate the plastic with water.
Begin scrubbing the plastic with the brush and bleach/detergent solution. Depending on the severity of the dirt and grime, you might find yourself having to rinse and repeat this step several times. Continue until the bucket chairs (or other furniture) is clean. Leave the solution on the chairs for at least 10 minutes.
Thoroughly rinse the bucket chairs of the bleach solution.
Wipe the chairs so that they are dry and carefully move the chairs back to their desired location. If you used a tarp, carefully tilt it so that the water sinks back into the dirt. Allow it to dry, then fold it and put away. Dispose of the cleaning solution in a sink or toilet. (The chemicals will likely kill any plant that it comes into contact with, so do not just dump it onto your grass.)