Aside from pointing a camera at your Gameboy, you can imagine it's pretty tricky to record quality GBA footage for your speedruns. The most common solution is to use the Gameboy Player attachment on a Gamecube so you can record the Gamecube's footage just like any other console. There are a variety of solutions to get this working, and to start I'll share a link to Tupto's guide on SRC that has some great information in it: https://www.speedrun.com/fusion/guide/wf5aa
Your two main options for using the Gameboy Player are to either use the official Disc that comes with it, or running a software called Gameboy Interface (a.k.a. GBI). More info on GBI can be found here: https://www.gc-forever.com/wiki/index.php?title=Game_Boy_Interface.
The Disc is nice because it "just works", and if you're lucky enough to have the disc then that's all you need. The major downside to using the disc is that it's got some input lag which may affect your gameplay.
GBI can be tricky because it requires some more steps to get set up, but once you get GBI running you should end up with a better experience with better looking video quality and reduced input lag. One way to get GBI running is to use an Action Replay. I personally don't have an Action Replay, so if anyone who does wants to update this page and replace this sentence with an actual description of how it works, that can be helpful, but for now I'll say I hear it should be as easy as just copying GBI onto the Action Replay and it boots right up.
Another way to get GBI running is to copy it into a regular Gamecube Memory Card along with a game save exploit that tricks a regular Gamecube game into loading GBI. To do this, you will need a Wii that has homebrew installed, and I've had success following the guide in this video https://youtu.be/Y3jq8dVe2ug. Once your Wii has homebrew, use it to install Gamecube Memory Manager, a.k.a. GCMM, found here: http://wiibrew.org/wiki/GCMM. Now that you have GCMM, use it to copy over your game save exploit and GBI from an SD card over to a regular Gamecube Memory Card, and now that Memory Card can be used to trick whichever game you chose to boot into GBI.
If you're starting completely fresh, the Action Replay is very likely the easiest way to get going. In my situation I already had a Wii and extra memory cards and a gamecube game that has a save exploit for it, so it was cheaper for me to use everything I had instead of buying an Action Replay. Whichever option you choose should get you running on a console and ready to speedrun with recorded footage.