This was very easy and only took me about 2 hours. I could tailor it a little better but this is good for now.
I took the whole chair apart and threw the arms away. They always get in my way anyway. Starting with the seat, I simply wrapped the vinyl around and hammered cut tacks (they are little nails that are called that; you can find them at any hardware store) on the straight sides. I had to use cut tacks rather than a staple gun because the chair is plastic and the gun I have wasn't strong enough to penetrate it.
When I got to the corners I twisted the vinyl like this as I tacked it. I like the effect it gives.
The top was difficult because of the adjustment knob on the back. I couldn't get the hardware off to put the fabric underneath it AND I had to contend with the metal "spine" that holds the backpiece in place so I decided to just make a slipcover for the top.
My solution was to make a pillowcase-like top piece and put velcro on the open end so that I could close it around the metal "spine".
And there you have it!