How to remove your Minecraft account
If you are not on the new Mojang accounts, you cannot delete your account unless you migrate it first on the Mojang Accounts signup page.
Once you are on the new Mojang accounts, go to account.mojang.com and sign in.
Under “My Account” click “View account settings”. Once you are at the settings page, click “Learn how” under “Delete Account”, and then click “Request Account Deletion”.
They will send you an email. Click the link in the email, answer the security questions and password and then click delete account.
Thanks to Jacob for this info!
Helpful Trick: Removing online data
Helpful Trick: Remove third-party logins
If you signed up for Minecraft using a third-party login such as Facebook or Twitter, you probably want to revoke all access from Minecraft. Click here to find out how.