How do you use HDMI with the GameCube?

Apart from internal modification (or through the use of upscalers), the Gamecube does not natively support HDMI. It will support 480p at most through the use of a component cable.

Reference: http://en.wikipedia.org/wiki/List_of_Nintendo_GameCube_games_with_480p_and_16:9_support

and also personal experience.

The official Nintendo Gamecube component cables were discontinued and are extremely hard to find (not to mention expensive when you do come across one). Third-party cables may do the trick; I use a third-party set for my Wii.

This answer will cover the following topics:

1. Why original GameCube component video cables are so expensive.
2. What are the alternatives for getting better video from a GameCube?

Cost and the Reasons for It

Original Nintendo GameCube component video cables are running between $150 and $200 on eBay right now. There are 3 reasons for this:

1. There were never very many cables made. In the U.S. the cables were only available directly from Nintendo and sales were poor enough that in the subsequent revisions of the console, Nintendo removed the port for the cable to cut costs.
2. The cable uses a custom connector that doesn't match any of Nintendo's other connectors. A Wii component video cable will not work wit a GameCube.
3. The cable contains custom integrated circuits that perform the conversion from the Digital A/V port's signals to component video.
4. With retro gaming garnering more and more interest and High-Def televisions being extremely cheap now, demand for the cables has spiked. Three years ago, you could find one for $85. (I can verify this because that was the price I paid for mine.)

GameCube Video Output Alternatives

You can get a European PAL GameCube, a SCART output cable that plugs into the normal A/V port, and another cable that converts SCART to component video. In NTSC (US and Japan) GameCubes the circuitry for this isn't there, so a SCART cable won't help. There are also some caveats to going this route. One being that the video quality is not as good as the official component video cable. See RetroRGB - GameCube Output Comparison for a good discussion of this.

There are a number of people working on the GameCube video problem. It appears that it's possible to get HDMI or VGA output on a GameCube for about $45 in parts.

There is an Open Source project that has been successfully used to output video from the GameCube at ikorb/gcvideo: GameCube Digital AV converter - GitHub.

On OSHPark, a site where you can buy unpopulated circuit boards, there are 2 projects for implementing High-Def GameCube video:

1. OSH Park ~ GCVideo Lite 0.9
2. OSH Park ~ Shuriken Video V2

The Shuriken Video V2 with the Xilinx xcs200a chip seems to be the best option. Installation requires good soldering skills and some modifications to the case, if you can find someone who has already assembled and programmed the boards. The components are surface mount devices, so you will need the right tools and excellent soldering skills if you must populate the circuit board yourself.

Apparently, a development board called the Pluto-IIx HDMI can also be used to do this. I hear that installation is more difficult than the Shuriken.

madman32ish - YouTube has done a number of videos on GameCube video output clones that are worth checking out to better understand some of this and see actual working examples.

Badass Consoles said it is working on a commercially produced version of the GCVideo as well. See GameCube - GCVideo - BadAss Consoles. This was just announced 1/18/2016 and is not available yet.