You cannot delete your account. Accounts are linked to games played and to various reports (including abuse reports) and we cannot remove your name from these records for technical and administrative reasons.
If you want a different name, please sign up again and get the name you want.
If your concern is having your email address in our system, and you never want to receive mail from us you can disable all mail in two ways: click on the unsubscribe link in any marketing message we send you, or when online, click on your username (top right) >> Settings >> Email Notifications and select to "Unsubscribe from all ChessCube emails"
Additionally, if you no longer want to use your account, you can simply not log on with that account any more.