The plot of Hamlet shows him moving through various stages or steps in his attempts to take revenge--the play, the stabbing, etc. These are the outward manifestations of his promise to his father.
But it's worth looking at his state of mind as the source of his actions. Hamlet doesn't appear to plan his revenge. He delays, prevaricates, philosophises, questions. There has been much debate as to the reasons for his delay--he has been presented on stage as indecisive, and his comment 'conscience doth make cowards of us all' has been interpreted as fear of the act of killing. But conscience in Shakespeare's time also meant 'consciousness', implying his endless self-awareness and inner debates preclude action. Yet, he is impulsive and kills Polonius without guilt ('lug the guts' into the other room) and arranges the death of Rosencrantz and Guildenstern ('they are not near my conscience').
It is likely that, while Hamlet is in a state of melancholy or depression, he can't do anything--he suffers a sort of paralysis. When he returns from England he has changed, is the energetic, active prince who is able finally to take action.