He specifically asks Horatio to:
* "Report me and my cause aright" (Act 5, Scene 2, Line 354)
* "Speak of me as I am" (Act 5, Scene 2, Line 356)
Hamlet wants his story and the events leading to his death to be known accurately, without embellishment or distortion. He wants Horatio to be his voice and to ensure that the truth prevails even after his death.