How can I turn a mattress into a couch?

Not sure about how to build it into that couch in particular. You'd need to build a frame for the mattress, plunk the mattress inside of the frame, and big square foam cushions on the side. This is a http://desiretoinspire.blogspot.com/2009/03/kims-komfy-kouch.html for a similar idea - I think you could take these plans and tweak them into a version of the DWR couch.

The problem with this is that a sofa made out of a twin bed is too deep to sit straight up on. You would need REALLY thick back pillows to not have to unprofessionally lounge around. Anyway, it looks like what you would need to do is build wooden boxes for the back and side, screw them together, cover that with spray glue and foam, then slipcover those, the bed, and whatever pillows you find/make. You could screw the bedframe to the back and side boxes, I'm sure. If you have a sewing machine, square slipcovers are really easy. Just do it all inside out with a lot of pins. The real one seems to be on some sort of platform base, I would skip that if I had to do it. On preview, I'm saying to not do anything to the bed itself, just to make a stiff rectangular fitted bedspread.

I think the bedspread slipcover part should have a top, front and left side. (The front and side would reach to the floor) and then on the back (the far long side of the mattress) and right side there should be a maybe 6 inch strip that velcroes onto the back and side box. Damn. I am completely out of things in my house to reupholster. Now I want to re-reupholster something.

Lots and lots of thick pillows.

http://www.daybeds.com/daybeds/upholstered-daybeds/3+18+4294848197.cfm Buy a daybed frame, so you can use the twin mattress and make it into a couch. See some upholstered examples at www.daybeds.com

I can tell you right now that your mattress is way too soft for that. Best bet, as mentioned, would be to put it on a daybed frame, but I'm not sure if a regular twin mattress would fit a standard one.

Well, I suppose you could make two boxes, one for the back, one for the end, out of mdf, and a platform like the diy couch without the wheels (boy, that's a great sofa bed!) Or hide casters up inside the frame so your sofa is very near the floor and the casters don't show. Fasten the whole thing together with L brackets so you can get it through the door when you need to move. Hinge the top of the boxes to stow blankets, pillows when they're not in use. Upholster boxes and frame. Slipcover the mattress. If you cover the frame and made a separate removable mattress cover (top and sides), you'd have something close, but not exact. You'll need lots of large pillows on the back to make it useful to sit on, perhaps pillows slipped into cases for day use.