Whisk together ketchup, brown sugar, garlic, vinegar, Worcestershire sauce, paprika and 1/4 cup water in a small pot. Bring to a boil, turn down heat, and simmer, stirring, until thickened, 10 to 15 minutes.
Preheat grill to medium, then oil grates. Before grilling, make a slit lengthwise in each hot dog. Grill hot dogs approximately 3 minutes; stuff each slit with a few pieces of cheese. Finish grilling, cheese side up, until cheese is melted, about 3 minutes.
Place hot dogs in buns and top with barbecue sauce and potato chips.
