You can re-use chicken bones for more broth if you don't overcook the chicken to start. Use the chicken skin too.
I like to use a Chinese chicken-poaching method I think is called 'white cooked chicken'to make a soup broth and a cooked chicken at the same time. Then I debone the chicken and use the bones and skin to either make another broth or enrich the first one further.
Submerge a small (no more than 3lbs) chicken in cold water in a deep pot. Put on a lid and bring it to a boil. Turn off the heat but leave the lid on. Set a timer for 30 minutes.
After 30 minutes the chicken is cooked. Separate the meat from the bones and skin and serve it in the broth or with a sauce over noodles (dandan sauce is nice) or in chicken salad, enchiladas, chicken pot pie, etc.
If you're making soup, you'll want to season the initial broth while deboning the chicken, for example by salting the broth and then simmering some garlic and ginger or a chopped carrot and a bay leaf in the broth.
Save the skin and bones from the chicken as you're stripping the meat. You can either return them back to the original broth to simmer longer and make it richer, or place them in a new pot with cold water just to cover, put on a lid, and simmer overnight. This will yield a second delicate but still flavorful broth.
I freeze all my broth unsalted so I can later combine it with more heavily salted braising liquids to make a nicely balanced soup.