There is no local play. The only way to do this over LAN or Wifi is to hack your Wii. It's relatively simple and the only way to get continued support for this discontinued product.
1. Hack your Wii with LetterBomb
- On Computer: If there is a folder titled "private" on your SD card,
rename it to "privateold"
- On Computer: Copy the "private" directory from the LetterBomb
download to the root of your SD card.
- On Computer: Take your homebrew (preferably Hackmii Installer) and
put it in the ROOT of your SD card as "boot.elf". Note that you can
download the Hackmii Installer directly from the download link
provided above. You will be asked for U (American), E (Europe), J
(Japanese) or K (Korean).
- Wii: Put your SD card in your Wii and turn it on.
- Wii: Go into the Wii Message Board and navigate to "Today",
"Yesterday" or "Two days ago".
- Wii: Click on the appropriate envelope, sit back and prepare for the
2. Patch your game with Wiimmfi
You will need to check that link to get additional links and files. Here's a simplified version of what you'll be doing.
Patching disk images
You will need:
- A (soft)modded Nintendo Wii
- The Homebrew Channel and
- A Custom IOS (cIOS) to change the way your Wii accesses data
- USB Loader GX (or a different USB loader)
- A USB storage device (eg. HDD, USB stick) formatted for use with USB Loader GX
- Wii Backup Manager or a different WBFS Manager.
Follow the directions and you'll be good to go. Make sure you follow all the steps. If you mess up the part with custom IOS you're going to brick your Wii. As long as you're paying attention and following the steps you'll be fine. It's literally one of the easiest consoles to hack.