1) Download the LoL client installer from here.
2) Launch the installer and follow the prompts in the setup wizard. This wizard will install the basic patcher for LoL.
3) Launch League of Legends and the patcher will complete the installation of the game client.
4) Press the Launch button at the top of the patcher screen and the game client will open.
5) The game client will continue to download the rest of the game contents. You may log in to the client during this time but will be unable to start a game until the download is complete.