Can I brine chicken in buttermilk after I boil the chicken?
Short answer: Brining after boiling won't do much (if anything), and you probably don't want to boil all parts of the chicken anyway.

One primary purpose of brining is to hydrate the cells of the muscle tissue before cooking, which allows the cells to hold on to more moisture during the cooking process via denaturation of the proteins.  If you cook the meat first, then the heat will permanently denature the proteins first, so the brine will essentially be useless.

The reason why you want to pre-cook chicken wings is that (a) they have a lot of fat which melts off into the cooking liquid, and (b) they have a good amount of connective tissue (collagen) which needs to be cooked low and slow to soften into delicious gelatin.  Most of the other parts of the chicken don't have nearly as much fat and collagen, so boiling them would likely only serve to overcook them.