Either direct me to where I can get a recording system for DS, or help me set up emulation. Thanks.

Because of the sketchiness of how emulation works, I can't post links on how to distribute/obtain ROMs here, but most people just use DesMuMe or another DS emulator for purposes of emulating DS. Alas I can't really help on getting that set up. Also DS emulation is tough on the computer and requires decent specs, especially if you're recording, so if you have an older or weaker computer you may experience other issues.

For DS capture, I got mine from here: http://3dscapture.com/ds/ . It cost 53$ for the card which you would have to install yourself, 75$ to have it installed to your DS (I believe you need an original model, I dont think you can use this for a DSlite), or 135$ to get a DS with it pre-installed sent to you. This site ( 3dscapture.com ) also has a place to order a 3DS capture which allows you to get a capture card installed to a 3DS or a 2DS, which is a bit more expensive but allows you to record both DS and 3DS games.

For recording purposes or streaming purposes, OBS (Open Broadcast Software) is the best. There's tons of resources out there for on how to use the software through a quick google search.

My idea for recording gameplay from my DS Lite and eventually 3DS when I get another one, is to put the screen in front of my laptop's webcam and look at the screen of the DS on a monitor I'll have connected to the computer to my right, and look at the chat which will be on a tablet to my left for when I stream that. Cheaper than a DS/3DS capture card.