What is the exact error you're receiving? Is it something along the lines of "Launcher has experienced an error. Please restart"?
If it is, then please firstly make sure you have no modified skin files, and if you do restore the originals. This causes issue with patching.
Secondly, try the following:
1) Uninstall LoL
2) Delete any traces left behind of Riot Games/League of Legends (look for folders in C:\ and program files)
3) Download CCleaner, run the registry scan, and clean up everything
4) Reinstall the game.
Let me know if you have any luck, or if it's a different error please attach a screenshot or tell us the EXACT message. Thanks!