A step by step procedure has been published in the League of Legends forum at http://forums.na.leagueoflegends.com/board/showthread.php?t=1946188&page=30
- Go into Ubuntu software center and look up wine.
- Install a program called winetricks and all the add-ons.
- Once installed look up a program called PlayOnLinux and install it.
- Open up PlayOnLinux then and Hit the Install a program on the right side of it.
- Once that is open go up on the upper menu and hit games
- Deselect commercial and select testing.
- Scroll down and select League of legends
When you are on the last installation page deselect the box to auto run after finish
Once it is installed and everything is finished bring back up the PlayOnLinux program and run it.