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Could any Riot Staff please sticky this forum so that more Linux users can benefit from this easy method?
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Before I start, YES, I did get a lot of help from other forum posts. However, I found there was no single forum post that effectively led me all the way through. I had to figure out a lot of this by myself.
Working as of: 25 March 2012
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The following is a VERY EASY guide on installing and running LoL on Linux.
(I will be using Ubuntu, but if you modify the code a little bit, it will work for any distro)
Step 1: Install Wine1.4 and Winetricks
code: sudo add-apt-repository ppa:ubuntu-wine/ppa
(adds ppa repository of wine)
code: sudo apt-get update
(updates the database/list)
code: sudo apt-get install wine1.3
(installs wine1.4)
code: sudo apt-get install winetricks
(installs winetricks)
Step 2: Install Winetricks Extra Components
code: winetricks vcrun2005
(installs visual c++ 2005)
code: winetricks wininet
(installs inet)
code: winetricks ie6
(installs internet explorer 6)
code: winetricks d3dx9
(installs directx 9)
code: winetricks corefonts
(installs several windows fonts)
code: winetricks adobeair
(installs adobe air)
Step 3: Change Wine1.4 Settings to "Windows 7"
code: winecfg
(opens wine config)
Then, change the Windows Version Setting to "Windows 7"
Step 4: Move a Previous Install of LoL
You should place the folder (Riot Games) into the following location:
(/home/[USER]/.wine/dosdevices/c:/Program Files/)
Make sure you replace [USER] with the correct name.
ALSO, .wine is sometimes hidden from view, so you may have to find it.
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Now that you have completed the above tasks, you will now have to open the LoL Launcher. To do that, you will have to type the following two lines of code into a terminal every time you want to play LoL.
code: cd "/home/[USER]/.wine/dosdevices/c:/Program Files/Riot Games/League of Legends/RADS/system"
(changes directory to where you have LoL folder installed)
code: WINEDEBUG=+ntdll wine "rads_user_kernel.exe" run lol_launcher $(ls ../projects/lol_launcher/releases/) LoLLauncher.exe
(runs program under debug mode)
(make sure to replace [USER] with the correct name)
The League of Legends launcher should now appear and begin updating.
CONGRATULATIONS on your new install of League of Legends on Linux!
IF this worked for you, please LIKE this, so that other people will be able to see this post.
IF it didn't work for you, or you have any questions, please feel free to leave a comment. 
Known Bugs: Because this is NOT meant to be played on Linux, there ARE bugs.
-Store in Lobby/Launcher (NOT IN-GAME) does not work. If you press it, LoL Launcher.exe crashes.
-Health Bar on HUD remains full, no matter how much health you have.
-Recall Bar remains full while you recall.