Basically, edit the html tag to include the face book namespace:
<html xmlns="http://www.w3.org/1999/xhtml" xmlns:fb="http://ogp.me/ns/fb#">
and add the open graph meta tags for title, description, and image to your page:
<meta property="og:title" content="Title that will show on FaceBook" />
<meta property="og:description" content="Description that will show on facebook" />
<meta property="og:image" content="http://yourdomain.com/images/icon.jpg" />
for the like, you should use a full path to the index page, instead of just the domain:
http://mydomian.com/index.php