I was about to post a message referring to super-thin LED TVs and the like, but once I watched about 5 seconds in the middle of the video I realized that, yes, the PS Eye DOES have that flat part at the back of the base that should line up perfectly with the TV. I guess you should only worry about spending the $6-7 (it's gone down in price quite a bit since January) if you don't want to get the precious bevel on your TV slightly scuffed. ;)
@Emmitt--I don't think this method would scuff the TV unless you knocked the camera around. And any clamp could do way more scuffing than the camera.
I think the only problem with this would be if the TV was up higher, mounted to the wall, etc. where the camera didn't capture the player with a straight on view. But better to try it first then get the mount if needed because you can angle it with the mount. If it's on a high table/platform there should be enough room in front of the TV without blocking the screen. @Hassan Great tip