How do you get Hamlet sent away on the ship?

In order for Hamlet to be sent away on the ship, you need to follow the unaltered timeline. Do the play about the murder, Hamlet kills Polonius, Claudius and Hamlet are still alive Saturday morning. Claudius will send Hamlet, Rosencrantz, and Guildenstern to England.

If you want to get on the ship yourself, you need to get Hamlet sent to England first. When he comes back he will tell you about the letter, and you can use that as hearsay in your next timeline. Talk to Claudius after the announcement of Polonius' death and before Hamlet departs and tell him the ship plan fails. He'll send you on the ship with Hamlet.