If, for example, my turn starts by playing a King's Court and choosing Conspirator, how many cards will I draw?
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: