How does Conspirator work with Throne Room / King's Court?
This is explained in the Intrigue rules


  Conspirator – You evaluate whether or not Conspirator gives you +1 Card and +1
  Action when you play it. If later in the turn you play more Action cards, you do not go
  back and reevaluate a Conspirator played earlier. 
  
  For the purposes of counting actions,
  if you Throne Room an Action, that's one Action for the Throne Room, one for the
  selected Action played the first time, and one for the selected Action played the second
  time. For example, if you play Throne Room on Conspirator, the first Conspirator will
  be your second Action, and won't give you +1 Card or +1 Action, but the second
  Conspirator will be your third Action, and you will get +1 Card and +1 Action for that
  second Conspirator. Action - Victory cards are Actions


For your example, if you play King's Court and choose Conspirator, you would draw +2 cards and get +2 actions.  The first copy of Conspirator would count as your second action, and then the second and third plays would qualify.

The sequence would be:


Play King's Court (Action #1)
Choose Conspirator (Choosing does not count as playing an action)
Play 1st Conspirator (Action #2 - no bonus from Conspirator)
Play 2nd Conspirator (Action #3 - receive bonus from Conspirator).
Play 3nd Conspirator (Action #4 - receive bonus from Conspirator).