Create a slip knot with a 3-4" tail. Bring the yarn from the center of the ball around the hoop. Create a single crochet stitch around the hoop.
Continue making single crochet stitches around the hoop.
Once you reach the first stitch you made, join the last with a slip stitch to close the round.
You don't need to join rounds, simply continue to crochet into the next stitch no matter which round it is from. Working this way will make the fabric grow in a spiral fashion that won't show any seams.
Once the crochet cat cave has enough length, finish the last round with a few slip stitches to even out the end. Tie off the yarn and weave in the end.