Question: How can I take a screenshot of my PC's BIOS?

Answer: 
Take a camera, point it at the screen and take a picture.

Unless you have some kind of very fancy KVM switch that works over network (no, the cheap home user ones don't do that). Then you can do a direct screen capture on the receiving end, I think. Requires another computer, though.

Yet another option would involve a graphics card with TV out, a video capture card and another computer too. And some hope that video out works in text mode out of the box. I think you can guess where that goes.

Seriously, take a camera. It's by far the easiest and quickest way of getting a screen grab of something that happens way before you control the computer.
Some motherboards can be set up to push the boot-up and BIOS interface out the serial port. If your's can, then you could connect it to another machine with a terminal emulator running, and take a screen capture of the terminal emulator window.