First of, the things you need are:
--A with system version 3.2.
--A FAT/FAT32 formatted (no bigger than 2GB, as the Wii can't read SDHC cards).
--An (or a USB device that can act as one, such as a digital camera or digital photo frame).
--"" for Wii (random, I know, but you should really own this anyway).
--The "" (a modified Zelda save game that runs the Homebrew Channel Installer).
--The .
--"," a Homebrew app needed for the Wii to read DVDs from the drive.
--"," a Wii port of a media player that plays commercial DVDs.
There are essentially two main tasks here: installing the Homebrew Channel, and installing the DVD player. I'll start with the HBC.
1) Insert your SD card into your reader and copy the "private" directory of the Twilight Hack onto it. Also copy the Homebrew Channel "boot.dol" file.
2) Insert the SD card into your Wii and delete your Zelda save file in the Wii File Manager (start a new game if you don't already have one). If you care about your saved game, you can first copy it to your SD card and copy it back after you've installed HBC.
3) Before inserting Zelda into your Wii, flip it over and take note of the text on the inner ring. It will either read "RVL-RZDE-0A-0" or "RVL-RZDE-0A-2" -- both versions will work, but you need to make sure you apply the correct hack (RVL-RZDE-0A-0 requires TwilightHack0, RVL-RZDE-0A-2 requires TwilightHack2, both included in the package).
4) Switch to SD view in your Wii File Manager and copy over your correct version of the hack.
5) Insert your Zelda game disc and start the game. You should see the Twilight Hack save game. Select it to start the game. Once you can control Link, walk backwards (weird, I know) until the Homebrew Channel installer starts. After a minute or two, the HBC will be installed and can be accessed just like any other Channel on your Wii.
Once you have the Homebrew Channel up and running, you can now install such as , , ... even ! For now, here's how to get DVD playback on your Wii with MPlayer.
1) Pop your SD back into your card reader. You can erase the "boot.dol" file and the "private" directory if you'd like (we no longer need the Twilight Hack or Homebrew Channel installer as it updates automatically and can be removed just like any other Wii Channel).
2) Create a directory on your SD card and name it "apps" (no quotes). Copy the "dvdx" and "mplayer" directories into the "apps" folder.
3) Eject your SD card and insert it back into your Wii. Turn your console on and start the Homebrew Channel.
4) Now you should see the two apps you just copied onto your SD card. Run the DVDX installer app. When that starts up, select "Normal Installation" and wait for it to finish.
5) Go back to the Homebrew Channel, insert a DVD movie of your choosing into your Wii and start the MPlayer app. To play the movie, select "DVD-Video" from the MPlayer menu, and then "Play DVD" (or "Play Title #1"). After a few moments your movie will load and you'll be enjoying a feature that should have worked out of the box to begin with!
*I should mention that while this method does work, the MPlayer software is currently pretty buggy. DVD menus are pretty hit or miss and not every movie I tried even plays. Hopefully they update the MPlayer software at some point, but I suspect it was more a proof of concept. There's always a certain amount of risk with installing unofficial, third party software, but hey, that's part of the fun. In any case, the Homebrew Channel is a really cool piece of work and will definitely breathe new life into your neglected little Wii.