Depending on your native screen resolution the game might not be able to start up with the standard configuration it sets. In this case you can edit the config file of the game directly.
- Completely shutdown your game and client.
- Open Notepad as administrator.
- Do this by searching for Notepad in your Start Menu.
- Right click Notepad and start it as administrator.
- Select File > Open and navigate to C:\Riot Games\League of Legends\Config and open the file called data.cfg.
- Search for the lines Height and Width and edit them accordingly depending on your native screen resolution (eg. 1920×1080).
- Save and exit the file.
- Start your game.
If the game still fails to launch repeat the above steps and also edit the line WindowMode from 0 to 1.