I am not sure what is considered 'best' here but I'll give 3 methods to capture DS games.
Wii U VC if the game is available there. (Good quality + HDMI capture cards aren't that expensive)
DS Capture card (Good quality but quite expensive. I am not even sure if they're still available)
Or pointing a webcam / phone at your DS. (Best cheapest option to get started)
I'd like to know it too. My only option would be pointing a phone camera at my DS, but I don't know how to keep both screens visible and still be able to play without shaking my DS too much. I would play Resident Evil DS.
@WanderAgro How I fixed this is by building a stand from LEGO holding my DS up, this way it is easier to point my phone at both screens and without shaking the DS. Then I upgraded to the Kid Icarus Uprising Stand. Hope this will help!