Create your diluted bleach solution using approximately 1 part bleach to 20 parts water.
Submerge the bucket in the bleach/water solution and spin it around a few times until all surface areas have been soaked.
Lift the bucket out of the solution and use a sponge, cloth, or brush to wipe down the entire inside and outside of the bucket with the same bleach/water solution.
Rinse the bucket with clean, clear water at least three times to ensure that all of the bleach has been removed. Set the bucket upside-down to dry.
Notes
Be careful when using bleach - you should do this project in a well-ventilated area, wear gloves to protect your hands, and wear clothing that can get a bleach splash on it without damage.