In Shakespeare's Hamlet, sometimes what happens off stage, or has happened before the play begins, is as important as what takes place during the course of the visible action in the present. By what methods and how successfully does Shakespeare let us know about off-stage or previous events?

There are many ways that Shakespeare informs the audience about prior or offstage actions. The main one is through conversations between characters about those events. In Hamlet, there are many instances of this type of reporting. In this play, the supernatural also plays a huge role in telling what probably happened to old King Hamlet.
Two important reports of offstage events concern Hamlet's aborted trip to England. He comes onstage and explains that he had found a letter with instructions to kill him when he arrived. He swapped it out for one that said to kill his companions, Rosenkrantz and Guildenstern. He escaped the boat en route. The news of his escape was reported already in a letter to Horatio. Later at the play's end a messenger comes on with the news of their deaths.
Another major event that happens offstage is Ophelia's death. This is major because she is such an important character, and because Gertrude delivers the news to Laertes. She tells him Ophelia drowned after falling into a river. The other important aspect is that we never hear the story from anyone else, and Gertrude does not actually say she witnessed it. The extreme ambiguity of her account leaves the door open to interpret Ophelia's death as accident, suicide (as the gravediggers discuss later), or that someone else killed her.
The Ghost is problematic as a source, as Horatio warns Hamlet. What if he isn't real, or what if he's an evil spirit pretending to be the dead king? In any event, Hamlet believes him that Claudius killed him, and therefore swears revenge.