The seat has a hard backing and I just wrapped it like a present using a staple gun to secure everything on the bottom. Nice and tidy, and the staples don’t show.
The back was a lot harder. Since almost all of it can be seen, it had to look pretty. First I traced the outline onto the fabric, leaving 1/2-3/4 of an inch of seam allowance, cut it out, and set that piece to the side.