Shakespeare's Hamlet is just one of the many plays that is as pertinent to its audiences today as it was in the late 16th and early 17th Centuries. This is why the Bard's plays are timeless.
Act One is about a young man who returns home from college for his father's funeral. His mother has remarried—so soon after his father's funeral that Hamlet sarcastically notes that the leftovers from the funeral could have been used for the wedding celebration.
Thrift, thrift, Horatio. The funeral baked meats
Did coldly furnish forth the marriage tables. (I.ii.185-186)
Act One also includes a father (Polonius) sending his son (Laertes) off to college with advice as to how to best handle himself and avoid problems. Laertes gives advice to his sister (Ophelia) about keeping her distance from Hamlet for fear that his intentions are not honorable. These kinds of family dynamics take place in modern society.
It is also not unusual that a parent would remarry and the child (i.e., Hamlet) would feel resentment, as if the new spouse were taking the place of "missing" parent.
In Act Two, Polonius sends someone to spy on his son. News of international affairs is shared in this act. The concept of parents interested in a child's actions is repeated as Claudius and Gertrude decide to have Hamlet watched. All of these things can be seen today.
In Act Three, Claudius and Gertrude want Ophelia to see if she can find out what is wrong with Hamlet, so he believes that Ophelia has betrayed his best interests and he angrily rejects her.
Get thee to a nunnery! Why wouldst thou be a
breeder of sinners? (III.i.130-131)
Hamlet looks for proof that Claudius is (among other things) unworthy of his mother. After Hamlet tricks Claudius into revealing his guilt at the play, Claudius sends Hamlet to England, accompanied by his old school "friends"—ostensibly to get his stepson away from him. Hamlet also directly confronts his mother, accusing her of being unfaithful to his father's memory. These family dynamics are not unusual in contemporary society, especially in light of the frequency of blended marriages. Hamlet's anger with his mother is also a present-day theme.
In Act Four, Hamlet and Gertrude have begun to better understand each other, and Gertrude lies to her husband about Hamlet's sanity to protect him.
Mad as the sea and wind when both contend
Which is the mightier. (IV.i.7-8)
One parent lying to a new spouse would not be unusual in a modern blended family. Hamlet plays word games with Claudius, irritatingly. Hamlet admires Fortinbras' behavior in doing things he cannot do; Ophelia's heart is broken and she suffers a breakdown; and, Laertes wants revenge for how Hamlet has harmed his family. All of these things are also contemporary occurrences.
In Act Five, Laertes and Hamlet fight over Ophelia—funerals bring out the worst in people in any day and age. Hamlet recalls the joys of his youth with Yorick who played with him as a child. Laertes plans to get back at Hamlet in a dishonorable way—allowing himself to be manipulated by Claudius. Claudius' desire for power and his failure to value the lives of others brings about his own downfall and that of those around him. All of these issues are typical of human relationships years ago and still today. Hurtful behavior like Claudius' (without consideration of the murder) can destroy friendships, families and lives.
These things show that Hamlet is a timeless play.