My question is, if I have a website for my subscribers to log in and have access to unique content how can I make the website verify that the user is subscribed to the channel? And continue to do this verification, so that if the user unsubscribes from the channel, his account is disabled on the website?
You would need to look at Check User Subscription by Channel in the dev docs - you’ll follow the Twitch OAUTH process (Authentication process) for users to give you permission to check their subscription to specific channels.
As for continuing to do the verification, the OAUTH does not expire unless the user actively revokes the permission, so you could check their status at periodic times - that is dependent on your setup and how many users you are looking at handling.