While Hamlet does speak to Horatio throughout the play, he does not give him any specific instructions about what to do after he dies. However, Hamlet does imply that he would like Horatio to tell his story truthfully and to ensure that his enemies, Claudius and Laertes, are held accountable for their actions.
Specifically, Hamlet asks Horatio to:
* Tell his story to the world: "Report me and my cause aright to the unsatisfied" (Act 5, Scene 2).
* Ensure justice for his death: "And let me speak to the yet unknowing world how these things came about" (Act 5, Scene 2).
Ultimately, though, Hamlet leaves the decision of what to do after his death up to Horatio. He trusts Horatio to do what is right and to act in accordance with his wishes.