I had to ask my friend Google what »biochar« is - it is activated charcoal, ground to a dust.
No experience with chickens and lice (i have ducks), but charcoal and wood-ash are fundamentally different: Wood-ash contains a strong alkaline (Potassium Hydroxide) that may kill off the lice - i'm not sure. Powdered Carbon will just make your bird black. I doubt that it has any effect on the lice. It might also color the lice black.
Let's wait for the chicken experts to chime in, but i would get some permethrin or diatomic-earth if you don't want to use chemicals.
Just burn some wood…LOL! Thanks for the reply...that's sort of what I thought. I can't use permethrin because I have a barn cat that sleeps in the coop/run with the flock and it seems like it's toxic to cats. I am not a DE person, either. Hopefully I can find some wood ashes.