What sort of camera do I need for portrait-quality shots?

Are we talking SLR cameras or point-and-shoot cameras?

Many SLRs can take a shutter-release cable, which is what you will want. However, if you were thinking point-and-shoot, there are a few of them that take shutter-release cables as well. I can only think of the Canon G10 and G11 at the moment for point-and-shoot, but they are a "pro" level point-and-shoot (in a way).

As far as connecting to a monitor, just attach the RCA to mini 1/8" plug to the monitor.

It's really tough to do your own headshot. The camera will autofocus on you--but half the time it will not focus on your eyes, which need to be in really good focus. The tip of your nose will be in focus, etc. half the time.

http://www.flickr.com/photos/little_pooky/42611014...

http://www.flickr.com/photos/little_pooky/36017304...

http://www.flickr.com/photos/little_pooky/35169327...

http://www.flickr.com/photos/little_pooky/46307594...

To answer your question, I believe all Canon dSLR will tether to the computer. Some models will do it wirelessly, too (for a price). A second or so after you take a shot, the photo will appear on your screen.

Have someone else take your photos if it's important.

p.s. It drains so much power from the camera, by the way.

All canon EOS cameras with live view feature tethered shooting if you have a laptop. Ideal to preview the effect of your settings.

Something at the budget end like the Rebel XS or XSi with the cheap 50mm f1.8 lens will deliver excellent protraits if used with care. Shoot RAW. Shoot near a window with net curtains (improvised soft box and scrim)