we made a simple one out of wood.
No pictures, but it was pretty simple.
Length wise boards to bolt and old E30 seat to with crosses for support, then a board for the wheel to clamp to with a couple supports on either side... Pedals on the ground.
Worked well, however it needed longer boards to sit on as the seat put your weight almost directly over the very back of the base making it just barely stable.
As far as what to make it out of.... What you got?
Metal, wood and PVC are all popular choices, and I would go in that order when choosing what to work with.
If you can weld, metal would be heavy... but very awesome. If I had the room, and a welder I would basically just build a cockpit FSAE style (or contact a school that participates and try and buy and old frame, cut out the rear and weld in a support for the wheel and pedals.
Wood is pretty do-able for most people. We just used a pile of 2X4's and a couple of guys with no wood working experience. As I said, turned out OK, but if I did it again I would probably go MDF and make more of a "monocoque" then a frame. Wood does not look as cool as metal, especially for a race car... but it will function good.
PVC seems like it would be easy with little tools... but I can't help but think it would be weak and flexible. No real experience of any kind with PVC though.