Unfortunately, I did not keep very good records of what exactly I did. This is more of the general idea than a specific pattern.
I started out with the bottom. I like to use the four-chain ring to start rounds. I double-crocheted for a couple of rounds, then half-double crocheted the final round. You want your bottom to be a little smaller than the diameter of the tape core.
Next is the hard part. Single crochet the next round around the tape core. You will end up with big loops around the tape core. As annoying as this step is, it is necessary to cover up the ugly cardboard of the tape core. I did 2sc in one hdc, 1sc in next hdc, rep until end. This took me several attempts to get it right, and I ended up frogging it more times than I would have liked, but stubbornness won out.
Now, you could stop there, but why? I thought it looked a little plain, so I figured I'd play with patterns along the side. I did a 3dc cluster for the first round, because there were so many stitches. Then I did ch 1, 1sc, ch1, 2sc, *sk next st ch 2, 2sc, ch1, 2sc in next st (rep from *). Join last ch2 with first ch1. (This is a simple sc shell in case my directions aren't clear.) I did this for 2 rounds. For the last 2 rounds, I used a simple ch3 edging (ch4, sc in first ch2-sp, ch 3, sc in next ch2-sp). On the first edging round, I crocheted in the loops from joining the bottom to the tape core to keep the sides from flopping down.
This now makes a nifty little cup for holding pocket change or whatever.