Login to the Avocado web app at http://avocado.io/login
Visit your settings page (the last tab that looks like two little drawers)
Scroll to the bottom and look for the "Close account" link.
IMPORTANT: If you choose to delete your account, your partner will still have access all of the existing data that the two of you created until the time that they decide to also delete their account.
If you signed up for Avocado using a third-party login such as Facebook or Twitter, you probably want to revoke all access from Avocado. Click the icon below to find out how.