To make brine, in 3-quart saucepan, mix salt, sugar and chili powder. Stir in 1 1/2 cups of the water. Heat to boiling over medium heat, stirring frequently. Reduce heat to medium-low; cook 2 to 4 minutes, stirring frequently, until most of salt is dissolved. Remove from heat. Stir in remaining 3 cups water and the beer. Cool, stirring occasionally, until room temperature and all salt is dissolved, about 20 minutes.
Remove and discard neck and giblets from chicken. Rinse chicken with cold water. Place chicken in large resealable food-storage plastic bag. Add brine. Squeeze bag to remove excess air, bringing brine up around chicken to cover; seal bag. Set bag in medium bowl; refrigerate 5 to 6 hours to marinate.
Heat gas or charcoal grill for indirect grilling as directed in owner's manual. Remove chicken from brine; discard brine. Insert meat thermometer so tip is in thickest part of inside thigh and does not touch bone. Place chicken over unheated area on grill; cover grill. Cook with medium heat 1 1/4 to 1 3/4 hours or until thermometer reads 180°F and legs move easily when lifted or twisted.