First go to the character generation page. Then click the box next to Customization. This will open up the same options you'd find on your character creation screen. Proceed to building what your specific character looks like. Without special server privileges, the Armory won't release how they look, so it's a manual process. If you skip this step, the final avatar will be a blond human male wearing your armor.
Once your character is looking back at you, type in your character's name, region, and server. Click on submit. At this point you'll be given two links, one of your full body look (.../signature.jpg) and one of your portrait (.../avatar.jpg). What I do is copy the links into new web browser windows and the images update almost immediately. They're now ready for use in your favorite signature generator (the one I used at the bottom) or as your representative on all your favorite WoW community boards. Check out some examples over at Need More Rage.
Now, before any QQ'ing starts, be aware that the grumpy coders are still working out some of the bugs, the biggest one being that if the Armory's down, instead of your picture, you'll get an "Awaiting Armory..." message. This is a step up from just seeing your char nekkid like before, ok? /wink The full body shots are also fairly small and cloaks have yet to make an appearance. Plus, if you hide any elements in game, like your helm, there's no similar option to remove them from your avatar.
If you enjoy making artistic avatars, I don't see this replacing your skills, but recommend you consider it a potential tool. The overall artistic aspects are kind of flat unless you do something to dress them up. There isn't an option to change the backgrounds yet, either. For me, it's a way I can grab my current character's look online without the Model Viewer (aka while I'm at work), and do some fun things with it.
If you have any feedback or suggestions for improvement, be sure to hit up their GrumpyCoder forums. And if you do try WoW PaperIdol out, link us some char picts and tell us your thoughts in the comments!
[Special thanks to Team Ratshag!]