I have no idea if you'll see this since its so late. But I'll post anyway.
Amazon.com: StarTech.com PCI Express HD Video Capture Card 1080p - HDMI/DVI/VGA/Component TV Tuners/Video Capture (PEXHDCAP): Computers & Accessories
This should technically work for what you want. Unfortunately, that's as cheap as they get. And I'm not even sure if its a good card. Unless you are doing this for a Youtube video series thing or are willing to pony up the cash, you might be better off using a video camera pointed at the screen.
Is it professional looking or offer the best results? No. However, its the next best thing if your on a tight budget.
And before you ask, there is no such thing as HDMI to USB/Firewire cables. If they do exist, they are non-standard. USB 2.0 doesn't have enough bandwidth, and Firewire is probably pushing it too.