No -- Rake only cares if you begin the round grappling; if so, then you get two free attacks. That's it.
So I'm an awakened dire tiger or something that has taken the greater grapple feat (for whatever reason). This gives me two grapple attempts per round. Do I then get to rake twice?
Grapple 1 successful (move action), do damage with natural attack (bite) and rake (2 claws)
Grapple 2 successful (standard action), do damage with natural attack (bite) and rake (2 claws)
Is this correct?
The wording of the ability is unclear about this. It says you get the additional attacks under certain conditions, typically when you grapple a foe. Since you must begin the turn grappled, it's clear that "grapple a foe" doesn't mean "begin a grapple" it just means "make a successful grapple check." I would rule that you'd get to use the rake ability each time you made your grapple check, as long as you started the round grappling.
A Synthesist or an Eidolon can use its own version of the rake ability, which is even better. The text here reads "the eidolon receives these additional attacks each time it succeeds on a grapple check against the target," and no mention is made of needing to start the round grappled. So maybe synthesist is what you're looking for instead of Awakened dire tiger =]