Mix dry ingredients together in a small bowl and rub into the chicken, making sure to get the rub into the nooks and crannies. Let rub sit on the chicken for up to 12 hours in the refrigerator.
Preheat a grill to medium heat.
Place dark meat on the grill first because it has a longer cooking time than white meat. Grill until the internal temperature of the chicken is 180 degrees F for all parts except the breasts and is 165 degrees F for the breasts when measured with an instant-read thermometer. When chicken is almost finished grilling, baste it with a generous amount of barbecue sauce all over the meat. Save any leftover sauce for dipping.
Stonewall Barbecue Sauce:
1/2 cup butter
1/2 cup cider vinegar
1/2 cup ketchup
1/2 cup lemon juice
1/2 cup Worcestershire sauce
1 tablespoon hot sauce
Combine all ingredients in a saucepan and slowly bring to a boil over medium-high heat; then reduce the heat and simmer until the sauce begins to thicken.