I use Camtasia to do any capture which has the computer interface or a website on it, and am able to output at 1920 x 1080. One of the tricks to success is making sure that your computer graphics card is set to a high resolution.
Another thing you could do is "fake" the interface. Do a fullscreen capture of the still website, then add the motion elements in After Effects, as well as any mouse pointer motions you want. It's done all the time, but doesn't quite look as "live" as doing Camtasia capture with the real thing.
Joe Bourke
Owner/Creative Director
Bourke Media
http://www.bourkemedia.com