To completely remove your Google Hangouts account, you would need to delete your Google Account. You can read more about deleting your Google Account here: https://support.google.com/accounts/answer/32046?hl=en
You can also simply sign out of Hangouts and just not use it if deleting your account is out of the question. Most people go this route, because they do not want to lose their GMail and other features tied to the account.
If you are wanting to remove your account due to some kind of abuse or harassment, you can always block individuals as well: https://support.google.com/hangouts/answer/2945109?hl=en-GB&co=GENIE.Platform%3DAndroid