I have a DS XL and I would like to record speedruns. However I have no idea how to do so.

the easy way is to simply point a webcam at your screens. You can also buy a capture card for other DS's, a good place would be http://www.3dscapture.com/ there's also a page for ds itself.

What about ds lite? (That ds page says that it's not compatible, and any amazon searches I try just come up with a ds game.)

DS lite is also unable to use capture card modifications. You'll need to use DS original for a capture card, new 3ds or use a camcorder.