Preheat grill for medium heat and lightly oil the grate.
Rinse the chicken and dry with paper towels; rub vegetable oil over the entire chicken, inside and out. Pour liquid smoke flavoring into the can of beer and add the garlic clove. Set the chicken upright in a grill pan with the beer can inserted into the cavity. Season outside of skin with sea salt, black pepper, and garlic salt, gently rubbing the seasonings into the oiled chicken skin. Place the potato into the neck cavity to seal in steam and cover potato with the flap of neck skin. Secure the skin to the potato with a toothpick.
Place chicken upright on the hot grill and grill until chicken is no longer pink inside and the skin is crisp, about 2 hours. An instant-read meat thermometer inserted into the thickest part of a thigh, not touching bone, should read at least 165 degrees F (75 degrees C).