Generally, the best ingredients for any compost pile, including one within a chicken run, is "stuff I can get for free or need to get rid of anyway". Within reason, anyway, of course.What do you put on the chicken compost pile? Thinking adding a small one to the run of my gold deathlayers.
So, for carbons...have fall leaves? wood chips? Spent hay? Those work great.
Greens/Nitrogen? Food scraps from the kitchen. Scraps and weeds from the garden. Grass clippings from the yard.
As with any compost pile, you'll want more carbons than nitrogen. Typically, with a small pile in the run, you'll pile it up, they chickens will kick it apart and spread it out. So, that's the "turning" action you'd do in a typical, non-chicken run pile.