Brine the chicken: In a large bowl, mix salt, sugar and 1/2 cup warm water until dissolved.
Pour beer into the bowl and fill the rest of the way with water until chicken is completely covered. Refrigerate at least 2 hours, preferably all day.
Remove the chicken and discard brine. Pat chicken dry with paper towels.
Rub chicken all over with the BBQ spice rub and olive oil.
You will be grilling the chicken using indirect heat. Heat grill to medium-high, then turn off the center burner and reduce the side burners to medium.
Place the chicken skin side up on the indirect heat. Close grill and cook until the thickest part of the chicken registers 150˚F on a thermometer—about 45 minutes (start checking around 35-40 minutes).
Mix BBQ sauce with beer or apple cider vinegar to thin. Baste chicken with sauce every 5-10 minutes, starting around 25 minutes.
Move chicken to direct heat and place the skin side down to char the skin. Grill for about 5 minutes. Check temperature to make sure the chicken has reached 165˚F before removing from grill. Let rest 5-10 minutes
Serve with BBQ sauce