I use 4 totes, and Dawn Blue Dish Liquid. In the first 2 totes, I put plain water. In the third, I add just a tiny bit of vinegar (1/4 tsp per gallon of water). In the fourth is plain water. Pay attention to the temperature of the water. It should be slightly warmer than lukewarm, so the chicken will enjoy it, and it's more effective in soaking, and loosening the dirt. Fill them enough so the very top of the back of the chicken is out of the water when the chicken lays down.
Put the chicken in the first tote. The first time they may fuss at first, but they quickly settle down, and enjoy it, then they'll lay down in the water. Let them soak for at least 15 minutes. Rub the feathers around the vent area so that they're clean. Continue soaking if needed. When the vent area is clean, move the chicken out of the dirty water, into tote 2. Now put a little Dawn Blue Dish Liquid in your hand, and a little bit of water, and begin working it around on the chicken's back, wings, and keep working it down to the tail area, and vent. Use a SOFT toothbrush to clean their legs, feet, and toenails. Wash up the neck some, but not as high as the head.
When they are clean, move them to the next tub with the vinegar water, and rinse them good in the vinegar water. The vinegar does a couple things. It removes all soap residue so the feathers have a sheen, and it balances the pH back out, since the Dawn is a bit alkaline.
Next do the final rinse. Don't skip this step. It removes the vinegar water, further balancing the pH, and any remaining residues that would dull the sheen. Now depending on the weather, you can wrap them like a burrito, or if it's warm enough they won't get a chill, pat them dry. IF you want to blow dry them, be sure the dryer is held far enough away, so they don't get too hot, or burned.
Now for the face, and head. Use a washcloth, OR baby wipes. I use plain water on a washcloth, cleaning the comb, top of the head, sides of the face (careful of the ears, and eyes), wattles, beak, etc. Now take a little petroleum jelly, or bag balm, and begin rubbing it between your fingers, so it melts some, then begin rubbing it on the comb. Don't use much, since you don't want it to attract dirt, but you want it to liquefy into a very thin layer as you work it into the comb, wattles, beak, legs, feet, and toenails.
Be sure to replace the water in the first tote for each chicken, and in the rest as needed.