You'll need a strip of fur twice as wide as you want your bag, and about an inch taller. Plus a little extra for hems and seams.
Check the nap of the fur before cutting out

With the inside towards you, fold over the top about an inch and sew at the bottom of the flap, like a really wide hem. This will create the tube your draw-string will go through.

Fold your material in half lengthways with the hem still at the top, and the inside still facing you.

Sew the vertical seam, remembering not to sew shut the drawstring tube.

Before you finish the bag you need to cut out your extras from the felt. I went for ears and bottom paws. You could also make horns, tentacles or wings.

I folded my material in half to get symmetical paws, and ears that matched.

Rearange the bag so that the sea you sewed in step 3 is facing you. This is going to be the back of the bag.

Poke any feet/paws through the hole at the bottom, accounting for where they will be when the bag is turned right-side out, and sew the bottom shut.

This bit can be a bit fiddly. Felt and fur don't play well together!

If you have other monster parts, now is also the time to cut slits and attach them like the feet.

If making ears, you can fold over the top and sew together to give them a "cupped" look like animal ears.

You will also want to poke holes and attach the eyes. Remember the seam is on the back.

Thread your cord through the holes, and secure the ends. Then turn the back right-side out.

Ta-dah!

You might want to free any fur trapped in the stitching, with a needle or your fingers, to hide the joins.