Remove chicken from brine, rinse, then pat dry. Discard brine. Drain wood chips.
Spray chicken with olive oil spray (or brush lightly with olive oil), then sprinkle with rub mixture and literally rub it into the meat.
Fill an aluminum foil pouch with soaked wood chips, seal and poke a few holes in it to allow the smoke to come out. Start grill on HIGH, place 1 pouch of wood chips directly over the heat until you see smoke coming through the holes of the foil pouch.
Lower heat and keep temperature around 350°F. Move the foil pouch to the side of the grill. Add chicken, close lid and cook for 9-10 minutes, then flip chicken over, close lid, and grill another 9-10 minutes.
Baste chicken with bbq sauce and cook another 2-3 minutes (flipping once more), until chicken breasts have an internal temperature of 165°F and the drumsticks are between 165-180°F.
Remove chicken from grill and brush with additional bbq sauce if desired.