While a perfectly reasonable case can be made (as you know) that Hamlet's primary motivation throughout William Shakespeare's Hamlet is his morality, an equally strong case can be made that the only motivation Hamlet has for doing anything in this play is how he feels at the time. Emotions rather than morality or reason, then, are what drive Hamlet's actions.
We know that Hamlet is an emotional person the first time we meet him. He is dramatically mourning his father's death as well as his mother's "o'erhasty marriage" to Claudius. He is criticized by Claudius and his own mother for dragging on too long in his mourning, and his first soliloquy reflects his emotional plea to be able to quit this world (a constant theme for him).
Every major interaction in this play reflects his emotional decision-making, including his primary decision to avenge his father's murder. Hamlet's interactions with the Ghost are all emotional rather than rational. Unlike Horatio, Hamlet immediately believes it is his father's ghost and follows it, heedless of his friends' warnings. He listens to the Ghost's story and rashly promises to avenge his father's murder:
Yea, from the table of my memory
I'll wipe away all trivial fond records,
All saws of books, all forms, all pressures past,
That youth and observation copied there;
And thy commandment all alone shall live
Within the book and volume of my brain,
Unmix'd with baser matter: yes, by heaven!
Even Hamlet's choice to "put an antic disposition on" speaks more of emotion than reason or morality. His response to the Ghost is not moral or rational, but emotional and dramatic; this is exactly why he doubts himself (and is therefore so indecisive) so often throughout the play.
His relationship with Ophelia is based on emotion rather than reason, for he has to know, as Laertes says, that his choice of a wife will not be his to make. When he talks to her alone, he expresses all kinds of emotions: he is angry, protective, contrary, loving, suspicious, and scolding. It is obvious that his emotions control this interaction, and his emotions are all over the place. The result is a confused and heartbroken girl who kills herself soon after this encounter.
His interactions with Rosencrantz and Guildenstern during the course of the play are solely based on emotions. He is happy to see them, he is furious that they conspired against him with Claudius, and he gleefully sentences them to death with his pen.
Hamlet rashly stabs Polonius while venting his emotions on Gertrude.
Hamlet is moved by drama and assumes the same will be true for Claudius. One of the least rational things Hamlet does is decide to present a play and watch Claudius's face. Though ironically it gives him the final confirmation he needs to kill Claudius, it is a plan based solely on two people's emotions: his and Claudius's.
Hamlet's near-brawl in Ophelia's grave is full of emotion and gives Claudius exactly the ammunition he needs to prod Laertes into killing Hamlet for him. In the final scene, we applaud those emotions because Hamlet finally achieves his goal and kills Claudius, but we mourn what is lost because of the emotion-driven journey Hamlet took to get there.
Horatio is Hamlet's foil, even-keeled and rational; Hamlet knows he is driven by his emotions and seeks his friend's counsel--though he rarely heeds Horatio's advice. Though he does avenge his father's death, Hamlet's emotion-based choices do significant damage to everyone around him.