Giving a nod to the naysayers, I would not attempt this setup if you have only one computer in the house - the point about usage contention is well-taken. Other than that, it's quite feasible, relatively uncomplicated (particularly if you like fiddling with hardware) and doesn't require a lot of exotic gear. I'll start with the simplest setup, and work up in complexity.
To meet your split screen requirement, the monitor will need to have PIP capability. Your two sources (Cable + PC, DVD + PC, etc.) will correspond to two inputs on the monitor. For the PC, the standard 15-pin RGB connection is perfect, equal to component/ HDMI/DVI in quality. The video card will need to support the highest resolution the monitor can display. Audio should be routed through the receiver (unless the monitor has digital audio pass-through), but this means switching audio and video separately. This could be simplified with a remote which supports macros, however.
Going a couple steps further (which is the approach I took), you can handle most (if not all) of your video on the PC by using its DVD drive and installing a video capture or TV tuner card. These cards typically have multiple inputs, so cable/satellite/ VCR/etc. can be routed through the PC. With this setup, monitor/receiver switching is eliminated and PIP takes the form of windows. The added benefit or tuner/capture cards is that you can get "free Tivo". BTW, I'm sold on using broadcast HD, as opposed to paying for local channels on satellite - the video/audio quality is just as good or better theoretically (and actually) in many cases.
If you want to take this route, here are some basic suggestions for hardware:
-Fast CPU (at least 1.6GHz, but 3.0+ is better)
-Motherboard with built-in 5.1 digital audio (SPDIF)
-512MB RAM minimum
-Quiet, well-ventilated/cooled case
-Large hard drive (160GB plus)
-HD TV tuner card
-Video card compatible with tuner card
-Wireless keyboard and mouse (optical)
-Windows XP (MCE not necessary)
If this will be your second PC, you'd be best off with a home network and broadband Internet. Aside from the obvious benefits of web browsing, you'll appreciate having high-speed access for automatic "TV Guide" downloading. If you go the Tivo route, the PC should remain on round-the-clock. You might consider an A/V component type case, but these are more expensive and limit your hardware options somewhat.
As for the plasma monitor, be very aggressive with the screensaver and power management. If you are sensible, you need not be overly concerned about burn-in. I can't say enough for the sheer delight (not to mention, the lessened eyestrain) of having a huge screen for PC work. I actually encourage the kids to use it for their computer games.
Even though I've thrown out quite a bit of information, there are more nuts-and-bolts details to be covered. I'd be happy to provide such, and can also recommend the site byopvr.com.