In what ways are the circumstances today similar to or different than those of Shakespeare and his intended audience?
To begin with the most obvious answer to your question, Hamlet was probably based on an earlier work by another playwright, and ultimately on historical accounts. These influences are discussed in the introduction to the eNotes "Hamlet Summary and Study Guide" linked below.
Hamlet also takes up two themes that meant a great deal to the audiences of his time, and in a slightly differing way, to contemporary society. These themes are the balance between private and public justice, and the dilemma of imperfect sources for critically important information.
Hamlet is a private citizen of the state of Denmark. He has no warrant from any earthly authority to remove Claudius, by killing him or by any other means. The dramatist indicates to us clearly enough that Claudius is guilty of the murder of Hamlet's father, but as for the outward shape of the state, he appears to be an efficient and tolerated ruler. How can Hamlet presume to risk the safety of the state on a private quarrel, especially since as a Christian he should remember that vengeance properly lies in the hands of God alone? Moreover, being a private citizen, he must proceed in his revenge by devious means, which directly result in a series of deaths of the half-guilty and the entirely innocent -- Polonius, Ophelia, and by the end most of the cast, including Hamlet himself. The dilemma of reconciling personal justice and what is right or expedient in the larger social sphere is one that remains with us today, and will always remain as long as human beings live in societies.
In his quest for revenge, Hamlet is also forced to rely on the word of a source he cannot trust and has solid reasons for distrusting -- the alleged ghost of his father. This dilemma would have spoken strongly to Shakespeare's contemporaries, in an age of witchcraft trials and spectral evidence, but in a more general form it remains a live problem for us today. Hamlet has no means of proving the ghost is really what it says it is. It gives little if any evidence that can be checked by Hamlet himself, at least until Hamlet hits upon the expedient of the play that mimics the presumed murder. It might be a creation of Hamlet's own melancholy and imagination, or it might be an emissary of the Devil himself, trying to mislead Hamlet to Hell the way the Three Witches misled Macbeth, tempting him with just as much of the truth as he needs to be led into mortal sin. This problem of uncertain sourcing for seemingly vital information is again one that remains with us today. How many times have we seen "experts" on television, for instance, later revealed to be in someone's pay who will profit from the advice they have given, or simply shown to be frauds? We still have to evaluate the whispering of ghosts, and since it is easier to make a claim than to investigate one, we will always be faced with this problem.