You can only disable your account. Make sure you are logged in to your account. Then use the link above. It's not clear what happens to your personal details. At least your username seems to be stored forever, because you cannot reuse it to create a new account, even after a few years!
If you signed up for Twitch.tv using a third-party login such as Facebook or Twitter, you probably want to revoke all access from Twitch.tv. Click the icon below to find out how.