Sorry to hear you guys are having problems with this. Luckily, this is usually just a failed self-patcher issue that is fixed up rather easily.
The first and most common solution is to force the patcher to repatch itself. You can do this by deleting this folder:
C:\Riot Games\League of Legends\RADS\projects\lol_launcher
Many players also fixed this problem by disabling the UAC(Windows Vista and Win7 Only)
If that doesn't work, try doing the following:
Manually Updating/Installing the .Net Framework 3.5
Making sure you are running the game as an administrator.
a) Right-Click on the short-cut
b) Select the properties tab
c) Check the box labeled "Run as administrator"
Failing all of that, i would suggest sending in a support ticket to our Technical Team Here and they will be able to help you better in a one-on-one basis