Preheat an outdoor grill for medium-high heat, about 375 degrees F (190 degrees C). Mix the brown sugar, chili powder, paprika, dry mustard, salt, and ground black pepper in a small bowl. Place the half-full can of beer in the center of a plate.
Rinse chicken under cold running water. Discard giblets and neck from chicken; drain and pat dry. Fit whole chicken over the can of beer with the legs on the bottom; keep upright. Sprinkle 1 teaspoon of the seasoning mix into the top cavity of the chicken. The beer may foam up when the seasonings fall inside the can. Rub the remaining seasoning mix over the entire surface of the chicken.
Place the chicken, standing on the can, directly on the preheated grill. Close the lid and barbeque the chicken until no longer pink at the bone and the juices run clear, about 1 hour 15 minutes. An instant-read thermometer inserted into the thickest part of the thigh, near the bone should read 180 degrees F (82 degrees C). Remove the chicken from the grill and discard the beer can. Cover the chicken with a doubled sheet of aluminum foil, and allow to rest in a warm area for 10 minutes before slicing.