How to close a League of Legends account?

Contact support about it. You can choose to either fully delete or deactivate your account.

Account deletions will permanently delete all personal information associated with your account. You'll never be able to access the account again, and once it's done, there's no bringing it back.

Account deactivation will restrict access to your account, but all of your account's data will be untouched. No one will be able to log into the account while it is deactivated and your information will remain on our database. This process is reversible at any time upon request. Please note that account deactivation will NOT prevent your ranking from decaying.

For up-to-date instructions on how to do it, you should refer to Riot's support website, linked above for convenience.

The question and other answer are outdated. As of now, Novmeber 2016, you can request to delete your account and it can be fully deleted within 15 days:

Account deletions will permanently delete all the information associated to your account, including username, Summoner name, email address, and purchase history. The account will no longer exist and will never be accessible again. This process is irreversible.

To delete your account, follow the instructions mentioned in this link. You may also want to read the difference between account deactivation and deletion (both options are available) in the same link.