I suppose that making a 'bent out of a BMX bike is one way to go, although I think that there are other better options that others have tried. Easyracer used to publish a set of plans to build a Toureasy out of two old ten speeds - I built one using the plans in the late 90's and still use it on more or less a daily basis as my commuter - although I did have a hiatus for a couple years while I was able to ride a velomobile. Anyway, there are some online plans available to build these - not quite as complete as the printed plans, but probably usable. See http://www.recycledrecumbent.com/toc.htm. Also my own website has some other links to similar websites (see www.velomobiles.net, go to the Obtaining a VM page, and then scroll down if the intra-page links don't work for you. There is a lot there, so you may have to scroll for awhile).
Another approach that would work better than the BMX idea, but is simpler and faster than the Toureasy is to get ahold of an old Mixte frame and use the low part of the frame as a seat support, and the seat tube as the seat back brace, or at least a mounting point for sprint braces (the two struts that support the back of many recumbent seats).
Either way, a simple way to make a seat is to take two pieces of plywood - one for the back, and one for the base, and connect them with a piece of piano hinge. That way you can alter the angle the base makes with the back.
You add all your own stuff to it. It's a very reasonably priced way to get into a bent. I liked that design very much as the CG is extremely low which I love (after getting used to). And I didn't want to be lying back like in bed, but sitting almost upright. And the pedals are lower than the seat which is good for circulation.