I recently bought a Game Boy Player without the start-up disc and went through this process with frustratingly little guidance available online, so I figured I'd lay it all out to save anyone else the pain and hassle:
The Game Boy Player add-on for Nintendo's Gamecube console requires a special launcher disc (Game Boy Player Start-up Disc) to function, normally, and it costs a fair bit (somewhere around $75 at the time of this writing) to get a complete package including the hardware and software disc. However, you can buy the hardware without the disc for substantially less (more like $25 at the time of this writing), since it's thought to be useless on its own. There is homebrew--known as Game Boy Interface (GBI)--that can replace the functionality of the launcher disc, but most of the ways to launch the homebrew are fairly costly ($30-50), require hardware modifications to the console itself and/or require a launcher disc of their own that can get scratched/damaged/lost and then you're SOL again.
However, there are a number of exploits available for common games, including Wind Waker, Twilight Princess and Smash Bros, among others, which allow you to launch homebrew on an unmodified console, and these are easy to utilize if you have a softmodded Wii. Just fire up GCMM (GameCube Memory Manager) on the Wii's Homebrew Channel and flash one of the exploits to your memory card. Then, perform the exploit on the Gamecube to launch an unsigned application named 'boot.dol' from the memory card. I have a copy of Smash Bros Melee, so I used the appropriate Home Bros exploit for my disc revision. Once it's loaded on the Gamecube memory card, I just start the game, go to 'VS Mode' and then choose 'Name Entry' and it loads the homebrew application.
OK, great so far. If I flash the default GBI application to my memory card, it fires right up and I can see the Game Boy Advance BIOS animation. There are some nice video options I can change, like zoom, etc. but it's stuck in nasty, flickery, line-doubled 480i. If you want the nicest, cleanest, most stable image, everybody knows you want "240p"/non-interlaced mode.
Now, this is where most of the other guides online stop being helpful. They typically expect you to have an SD card interface or modchip and they recommend using Swiss to change/set your video mode to 240p. This causes a couple of problems, though: 1.) only one homebrew application named boot.dol can live on your memory card at any time (i.e., so you can't have Swiss and GBI on the same card, typically), and 2.) even if you somehow get another homebrew application onto your memory card (more on this in a bit), Swiss can't launch another homebrew application from the memory card and then adjust the video mode on-the-fly the way it can with programs launched from disc, SD card, etc.
The simplest solution is to just use the speedrunning variant of GBI (known as GBI-SR; gbi-sr.gci in the GBI package), which forces 240p (yay!) but doesn't expose the other handy video options available with the standard GBI (booo!). This lets us drop Swiss entirely from the equation for this task, but you might/should still want to have it available in your homebrew toolkit for other purposes, and we can accomplish this with a little hex editing.
Just open the gbi-sr.gci in your hex editor and right at the top you'll see 2 instances of the word 'boot.dol' (you might have to scroll down a bit to see the second instance, depending on how many lines your hex editor shows). You can change that name to anything else--I changed mine to 'boob.dol' because it's a simple change that keeps the same length. Save and exit, and you should now be able to flash both Swiss and GBI-SR to your memory card without conflicts. Now, when you fire the exploit, it will boot into Swiss and you can choose your 'boob.dol' or whatever from Swiss' list of launchable content on the memory card to play GBA games in glorious, stable 240p or you can use any of Swiss' other awesome/handy features on your native Gamecube disc-based games.