2 things it could be:
(From known problems and solutions)

Patcher problems
The patcher does not let me click the play button and it does not say offline (most likely you see swirly page loading symbols with no content).

Most likely cause Active scripting is disabled.
open up internet explorer. Go to tools -> options -> Security -> Active scripting (near the bottom) -> Enable -> OK. Restart the launcher.

or

You may be using an older version of the patcher. While most people have switched, last week or the week before a few people may have missed the announcement.

Please reinstall again using the installer at http://signup.leagueoflegends.com/download