I have the NTSC game, a GBA SP, and a GC with Gameboy player, but I'm not sure what the best way to record gameplay is. I was thinking using the GC and then just recording A/V on a capture card. Is this the easiest way or is there a better one?

Also, where can I get a Japanese copy of the game for the cheapest?

I use ezcap for capturing http://www.ezcap.tv/usb-video-capture/ezcap116-capture-card I ordered mine from that page in 2012 or so. Before then, I had used older easycap. But your post sounds like you already have a capture card, so I'm a little confused...

If you want as good quality as possible, I'm not sure what's the best setup...

And as for japanese copy, you can find some on ebay usually for a reasonable price.