Melt butter over medium heat in a medium saucepan. Add onions and garlic and cook, stirring, until onions begin to brown, about 7 minutes. Stir in 1 cup water, ketchup, celery, brown sugar, Worcestershire sauce and mustard. Season with salt and pepper to taste. Bring to a boil, then simmer, stirring, for 30 minutes. Stir in lemon juice. Set aside 1 cup sauce for a final basting.
Grill chicken, skin side down on an oiled rack set 5 to 6 inches over glowing coals, for 10 minutes, turning once. Grill for 10 minutes longer, turning once and basting with sauce, until juices run clear when skin is pierced with a paring knife and sauce is nicely browned but not burned (breasts may take up to 10 minutes longer). Transfer chicken to a platter and use a clean brush to baste well with reserved sauce.