I use the tub and either bird, dog or human shampoo--depends on what I have available and whether or not I am treating for mites/lice (then I use dog flea & tick shampoo). I've even used body wash. Some use dishwasing or clothes washing detergent, but that really strips their natural oils badly.
Wet the bird under running water and run your fingers through feathers to find any mud or manure clubs and make sure they get wet enough to soften. Add shampoo and gently rub in, add more water to spread the suds. Let stand in a damp tub for awhile for the shampoo to sort of soak into the feathers. If using flea & tick follow the directions, which usually vary from 5 to 10 minutes. You may want to use a fingernail brush on the soles of the feet (and on foot feathers).
Rinse the bird under running water to remove the soap--you can also use a bucket of water with vinegar added to help remove the soap from the feathers. For hard feathered birds adding some glycerin to the final rinse water adds a sheen.