I am looking for a method to check if a currently logged in user likes a particular website or not. I don't want this without asking permission from user through an application. Is this possible?
You cannot check if a user likes a particular page unless they have their likes available to the public (as per the Facebook privacy settings). You cannot query a specific page for a list of user that like it unless you have particular rights with that page (and are logged in).
See the FB Graph API docs for more information: Graph API