You could go about this in two ways as I see it:
1) One PC, with 5 video cards (you might be able to do this with less than 5 if they're dual head cards, mileage will be variable)
2) Thin clients - one normal pc (as the server) and 5 small, cheap pcs with a very small amount of memory, a network card, a graphics card and a processor (something like a really cheap microATX board). The server runs the applications, and the clients just run an X server for their touchscreens.
Option 1 may not actually be an option, depending on how far the monitors have to be from the main PC - signal loss may become a problem, and you're limited by the number of expansion slots your server has. I'm not sure of the technicalities of having 5 input devices on one machine either.
Option 2 is obviously more expensive, but also more expandable - you can just add extra clients wherever you need them, and you could even do the networking with WiFi to eliminate wiring. The actual implementation is quite simple - X was designed to do just this.
Hope that gives you something to chew over.