How is Hamlet as relevant today as when it was written?
Firstly, it's a testament to Hamlet's relevancy that it is still read and performed today, almost constantly.
I think the major reason for why it is still relevant is in the character of Hamlet. Unlike some earlier dramas, where characters' actions were indecipherable, Hamlet is a fully modern man who does not act out of a prescribed code. In fact, he questions his surroundings and what is the best path of action. He also finds it very hard to seek revenge and is conflicted about his mother, his role in society, and what he should generally do.
This type of introspection is something we take for granted in characters in plays and films, but at the time it was remarkable. This openness about Hamlet's nature, and his questioning of himself, make him accessible even to modern audiences. Hamlet is a man of thought and words, rather then deeds, and many people can relate to that, especially in a complex modern world.