If your account had "Unverified" status after purchase, changing e-mail to your own would be the simplest thing.
Take a closer look at the "Account credentials" section of an e-mail You've received from GameStore.live. It has username and password to log in to your new account and also a server, so You won't get lost while trying to log in to different server.
Head to LoL official website, click on the "Login" button in the top right corner of the screen.
You will be redirected to a new page where You can use your credentials:
Insert username and password You have received by mail from GameStore.live, then check server, confirm your credentials.
In case You made everything correct, You will be redirected to Main LoL page again, but now, in the top right corner You will see "Summoner Name" of the account.
Move your mouse on it and click on the "settings" button, You will be redirected to "Account settings" page.
After getting inside "Account settings", you will see, that current mail status is "Unverified"
Click on the blue "Change Email" button. Then fill out the form needed. Insert your own mail, re-enter it and insert current password to this account.
The last step would be to check whether is that your mail shown and press "Verify" button if it is. That's all, account would be verified after doing that.