What plan does Hamlet reveal to the audience once he is alone on stage?

Hamlet gives a lengthy speech at the end of act 2, scene 2 when the significant act of the scene has concluded and he is left alone on stage. In this speech, Hamlet mourns his inability to do anything in defense of his father "upon whose . . . most dear life / A damned defeat was made." He curses himself for a "coward."
Hamlet continues in this vein for some time, but towards the end of his speech he reveals a plan, inspired by the performance of the actors he has just spoken to. He plans to have them put on a performance in front of King Claudius which will be "something like the murder of my father." He will then "observe [Claudius's] looks," stating that if Claudius should "blench" (go white, or otherwise react with obvious shock or disquiet) then Hamlet will "know [his] course." He is sure that Claudius's response to such a performance will cement whether or not Hamlet is right in pursuing a course of vengeance against Claudius for King Hamlet's murder.
The closing couplet of this soliloquy is extremely famous and encapsulates Hamlet's plan:

More relative than this: the play's the thing
Wherein I'll catch the conscience of the king.