Remove giblets from the inside of the chicken. Discard or save for another use. Rinse and pat the chicken dry.
Prepare grill for indirect heat - once the briquettes have enough ash on them, pile the briquettes to one side, leaving a "cool" side. If using a gas grill, turn on all burners until grill reaches 350 degrees, then turn off the burners on one side.
If cooking in an oven, preheat oven to 350 degrees. Line a shallow roasting pan with tin foil to catch any drips.
Empty out (or drink!) half of the can of beer. Place can in an upright chicken roaster. Carefully fit the chicken over the beer so that the can is snug within the cavity of the chicken.
Brush outside of chicken with olive oil.
Combine Cajun seasoning rub ingredients. Sprinkle seasoning all over the outside of the chicken. Pour any unused rub into an airtight jar for a later use.
Place chicken on the “cool” side of the grill or in the oven. Roast for 1-1/2 – 2 hours, rotating about halfway through cooking time, until an instant read thermometer measures an internal temperature of 180 degrees.
Remove chicken from the grill or oven. Allow to rest 15 minutes before serving.
While chicken is resting, bring the beer left in the can to a boil. Boil for 15 minutes. Mix with salsa and serve on the side as a dipping sauce.