How can I implement a visualization of archived convective data from numerical weather models?
The best model archive I know of offhand is Iowa State Meteorology's Gempak model data in their MTArchive.  Looks like model-wise, they have at least 10 years of GFS (and the predecessor, the AVN).  No idea if the models are complete files (include CAPE) or not, though.

Gempak software is free, though you won't get graphics nearly as polished as WxBell; they've spent a lot of time fine tuning their setup, and likely use newer more flexible tools (probably GrADS or OpenGrADS, both also free).  You could probably get GrADS to use the .gem files, perhaps by converting them?