It's really pretty simple. Hold the chicken firmly by the feet, test the water to be sure it's the right temperature, and wash away. The most complicated part is deciding which of 5 different shampoos and washing detergents is right for each bird.
Once a bird is washed, it gets rolled up in a towel to sit for a while.
We call this "chicken burrito." The birds love it. Really, they do. They're warm and cozy and have their buddies nearby. The towel picks up the excess water and makes it easy to hold the bird while I trim beaks and nails and clean their legs thoroughly. Dirt can get under the scales on their legs, even when they've been kept on shavings all their little lives.
Then they go into the "apartments" to dry for the night.
I do this in the house. Some people have an outside room to wash chickens. At my house it all happens in the kitchen and the den. I used to put the cage on the dining room table, but now it's a little more out of the way. For some breeds you might use a hair dryer to fluff them up. My birds are all supposed to have relatively tight feathers, so I let them dry naturally.
The white Old English bantams always have to be washed before a show. The Dominique bantams usually only get washed once in the show season. After that I can just wash their feet and legs. They're much easier to condition. I washed a bunch of them two weeks ago, before Paso Robles. It was a warm, sunny day, so they got to dry outside on the lawn.
The kitten is up on the table because he's "helping."