Purchase 5 D-rings at a fabric store. You can also use shower rings, solid key chains or other rings in a pinch.
Turn the bear rug over so that the felt is up and place a ring on one paw with the round part angled outward toward the claws.
Sew the D-ring on to the bear rug, using dental floss or some other strong thread. Drive the needle all the way into the backing, instead of just sewing through the felt.
Sew rings on the other three paws. Sew a fifth ring to the back of the head at the widest point.
Trace the bear rug onto butcher-block paper and put markings where the rings are. Cut out the paper form.
Hold the model up against the wall and position it where you want the bear rug to hang. Mark where each ring is with a pencil.
Drill screw hooks into the wall in the spots you marked. Mount each with the open end facing up.
Hang the bear rug on the screw hooks.