Update:
Checking if user has already liked a link
Checking if user has already liked a link is only possible when user has authorized you and you need an app to do so.
You can understand more about facebook authentication at facebook developers page https://developers.facebook.com/docs/facebook-login/getting-started-web/
After knowing that user is connected using facebook to your website, you can check the url_like table using query
SELECT user_id FROM url_like WHERE user_id = me() and url="Your_Page_URL"
A success response will mean that user has already liked your link.
Use iframeframe tracker jQuery plugin to detect when a user has cliked on Facebook Like iframe. You can assume that user has liked the page if a click happens inside the frame. https://github.com/finalclap/iframeTracker-jquery
And here is example with tracking facebook iframe http://www.finalclap.com/ressources/exemple/jquery-iframe-track/