Most good stories start with a fundamental list of ingredients: the initial situation, conflict, complication, climax, suspense, denouement, and conclusion. Great writers sometimes shake up the recipe and add some spice.
Mom just married Dad's brother. Also, war may be on the way.
Only a month after the old King of Denmark dies, his queen remarries – to his own brother. Hamlet is not happy to have his uncle as his new step-father. On the political front, Prince Fortinbras of Norway plans to invade Denmark.
Dad's ghost says mom's new husband knocked him off. Revenge!
A ghost shows up on the castle battlements, looking suspiciously like the recently deceased King. The ghost has a message for Hamlet: his father's death was no accident. Hamlet is supposed to exact revenge, which, when you're talking about the current King of Denmark and the husband of your mother, can be quite the conflict. Meanwhile, Polonius tells Ophelia, Hamlet's girl friend, to end whatever it is she's doing with Hamlet.
For reasons nobody really understands, months pass with no revenge.
Revenge theoretically shouldn't be too complicated, if you actually get it done. The complication comes when Hamlet doesn't get it done. All he does manage to do is go crazy, which is complicated in its own right, but more so when you're not sure if he's faking it or not.
The addition of the treacherous pseudo-friends (Rosencrantz and Guildenstern) complicates matters further, as spies tend to do. On the Ophelia front, she's no longer talking to Hamlet. When the former lovers finally meet, he berates her for having all those qualities that, according to him, all women possess (that would be deceit and treachery). Lastly, there's some strange sort of lie-detecting play that Hamlet has devised, which is supposedly going to prove whether or not King Claudius is guilty of murdering the former King.