Here's a fairly simple method to run homebrew on wii;
-a SD card, formatted in FAT.
-a computer with a SD card reader.
First make sure your wii Firmware is up to date. The latest system menu version atm is 4.2. (Note:I doubt you have one, but if you have a region converted Korean Wii, the update will brick it.)
If for whatever reasons, you can't connect your wii to the internet for updating the system menu, then dont give up, as there's other ways to run homebrew too. (eg. older versions of bannerbomb)
To boot homebrew code in your wii, you'll need the Bannerbomb exploit. You can download it from here. (for system menu 4.2)
There's a bunch of different versions there named something like "abd6a_v200.zip", the right one for you depends on your wii.
Make sure your SD card has no directory named "private", if it does, rename it. Then extract the bannerbomb archive to the root of the sd-card.
Now you could either choose beetween installing the homebrew channel, or going for the faster choice, of just running loadMii which is more crude, but doesn't require installing anything on the wii itself. (Though only some of the homebrew work without the homebrew channel.. WiiDoom does fortunately :) .)
So next download loadMii from here, and extract it to the root of your SD card.
you should now have the following in your card:
you probably dont need the 3 last files though.
Then download WiiDoom and extract it the root aswell, now you should have a prboom folder, and a apps folder. In the apps folder, you can find a wiidoom folder, which has the .dol file that you need to select later with loadMii to launch the game.
Copy any doom iwads to the \prboom folder.
Pwads go in the \prboom\pwads folder.
Once you're done with that, its time to try if it works :P
Insert the SD card to the front SD slot on the wii. The SD icon in the wii menu should light up. Then go to the "SD Card Menu" (not the data management one!) a message with "Load boot.dol/elf?" should pop-up. Click yes, and it should now boot the loadMii homebrew-loader.
You should get a ugly red screen with the directories and files of the SD-card. Browse to the \apps\wiidoom folder, and select the boot.dol file to launch WiiDoom.
Next you can choose the iwad, and one or more pwads to run with it.
I havent tried 0.4.2 yet, but with the old version, you had to unplug (if you had the nunchuck connected to the wiimote) and then plug the nunchuck again to the wiimote when the game loads, as otherwise it would not register any input from it.
The aiming with the wiimote is pretty damn hard at first, but once you get used to it, you can get pretty accurate with it (more than with analog sticks).
I suggest not using the wiimote in the air, but rather having something supporting under your hand, so that you can keep your hand in a steady position. The wiimote is REALLY sensitive to even the slightest moves in WiiDoom. Ofcourse you can aim offscreen, if you dont want to turn all the time.
So far I've tried Suspended in dusk, Plutonia 2, Eternall and Whispers of Satan. WoS had some memory issues though, savegames not working properly all the time. Suspended in dusk had one or two homs in a map or two, but worked fine otherwise.
Ofcourse I'm not responsible if you brick your wii(do I really need to mention this?) but this should work on 4.2.
I think if you use this loadMii application instead of installing the "Homebrew channel", then it shouldn't leave anything behind in the wii after you remove the SD card from the front slot.
Hmmm... does that invalidate the Wii in any way? My daughter would probably kill me if I screwed up.