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!