Re- arranging the scenes
For example you can open with middle, or end to achieve an effect similar to 500 days of summer.
The scenes which you remove have to be in the re-arrangement of the Hamlet plot.
What an interesting task! I can give you some suggestions on what to cut. It is actually fairly common in productions of the play to cut Hamlet's talking to the Captain in Fortinbras' army and his subsequent soliloquy, "How all occasions do inform against me." This scene connects to Hamlet's giving the kingdom over to Fortinbras in the end, but it isn't essential to the plot line.
While I hate to think that there are other scenes that aren't vital here are a few more that could probably be cut without a loss to the plot line of the play.
1. Polonius' advice to Laertes.
2. Laertes conversation with Ophelia.
3. Hamlet's request and the subsequent "performance" of the Aeneas' tale to Dido scene with the actors.
4. Some of the graveyard scene prior to Ophelia's actual funeral.
5. Most of the scenes with Rosencrantz and Guildenstern.
Scenes to add:
1. The actual scene of Hamlet with the Pirates on the boat.
2. The actual wedding of Claudius and Gertrude with Hamlet's behavior shown.
3. A loving scene between Hamlet and Ophelia to serve as contrast to his subsequent behavior in Act 2 and 3.
4. A conversation with Fortinbras.
5. More with Hamlet and Horatio interacting to further illustrate their friendship.
To think of more examples for either list just ask yourself -- what was kind of boring? What doesn't contribute to the final scene of the play? What minor characters and their actions don't contribute to a significant theme of the play? What kind of scenes did you like? What would you want more of? What do you "hear" about that you would like to actually see happen? Which character(s) was most interesting and could use another scene or two? Once you come up with your list, I think the rearrangement of the plot line will seem rather obvious in terms of chronology. If you play around with chronology, you have to ask yourself what is gained by doing it?