Preheat oven to 350°F. Grease and flour a 2-1/2-qt. ovenproof bowl. Prepare cake mix according to package directions. Pour batter into prepared bowl. Bake until a toothpick inserted into center comes out clean, 45 to 50 minutes. Transfer to a wire rack and cool for 15 minutes. Invert and cool completely.
Tint the can of vanilla frosting blue with food coloring. Spoon 1 cup of it into freezer-weight zip-top bag; set aside the rest. Set aside 3 tablespoons of the remaining cup of vanilla frosting; use a few drops of red food coloring to tint it pink. Spoon the rest into another bag.
Cut graham crackers with a serrated knife to make two 3-inch triangles for the ears. Cut two 3-inch slits on the top of the cake and insert graham crackers as the ears. Spread front of graham crackers with pink frosting.
Frost the cake blue. Use a fork to lightly pull the frosting outward to resemble cat fur. Snip a 1/4-inch corner from the frosting bags. Pipe on blue frosting to trim the ears and add more fur. Press Snow Puffs onto the front, close together, as the cheeks. Cut the marshmallow in half crosswise and attach as the eyes. Pipe a dot of vanilla frosting and add the Junior Mints. Pipe a white dot at edge of eyes. Pipe a few tufts of white frosting fur.
Soften the fruit chews in the microwave on high for no more than 3 seconds. Flatten one and shape into the tongue. Press the remaining fruit chews together and shape into the nose. Attach the nose and tongue with vanilla frosting. Insert strands of black licorice into each Snow Puff as the whiskers.