First, I chose the fabric from my stash that I wanted to use, and then I grabbed an old shirt from the goodwill pile (eg, not something I wanted to upcycle into a dress for the little one). I traced the lid on the old shirt, laid it wrong-side-up on top of the face-up fabric, pinned them together, and cut out the circle. You might be able to tell from my photo that I cut right on the line. What am I, sewing for the first time with no idea what I'm doing? It was late at night. But please note, leave yourself a seam allowance, dummy. Actually, I'd give it about a half-inch.
Once it's right side out, stick your hand in there to push the edges out all the way around, then iron flat.
The lid comes off of the hamper (it's stuck with a piece of velcro), which is nice for running it through the sewing machine. There is a piece of plastic boning that gives the circle its shape, and working with this was something I learned through trial and error. I think it's still helpful to pin the fabric to hold it in it's general place, but expect to do a lot of tugging.
To cover the handles, I first flattened down the body of the hamper (it's got velcro tabs to hold it that way). Then I cut out two rectangles of my fabric, approximately the same length as the handles (one for each side). I kept the fabric right-side-out and folded it around the handle, then sewed a straight line down the outside, making sure NOT to catch the actual handle in the seam.
There's the finished product! It still doesn't match my daughter's room, but what are you going to do? It lives in the closet anyway.