Place the beef brisket in a large slow cooker with the fat side up. Pour the coffee over the meat. Cook the brisket on LOW for 24 hours.
Meanwhile, stir together the ketchup, cola beverage, Worcestershire sauce, mustard, liquid smoke, and brown sugar in a bowl until well blended. Refrigerate until needed.
After 24 hours, remove and discard any fat from the brisket. Use a fork to pull apart and shred the meat. Pour the sauce over the meat, stirring to coat evenly, and cook 1 hour longer.