* To stay alive and tell his story: Hamlet tells Horatio, "If thou didst ever hold me in thy heart, / Absent thee from felicity awhile, / And in this harsh world draw thy breath in pain, / To tell my story." (Act 5, Scene 2) He wants Horatio to live so that he can tell everyone the truth about what happened and clear Hamlet's name.
* To report what he sees: Hamlet also tells Horatio to "look you how cheerfully my mother looks, / And for your love delight not to break / Into a passion of loud lament." (Act 5, Scene 2) He is asking Horatio to be a silent observer and not reveal the truth of Hamlet's death immediately.
* To tell the truth about the events: Ultimately, Hamlet wants Horatio to "report me and my cause aright / To the unsatisfied." (Act 5, Scene 2). He wants Horatio to be his voice and tell everyone what truly happened.
Hamlet's requests are about preserving his legacy and ensuring that the truth is known. He trusts Horatio to be honest and to carry out his final wishes.