Why is Hamlet considered to be a representation of modern human? How are his problems and preoccupations universal?
The answer to this question is the reason why Hamlet has stood the test of time for over 400 years. There has never been a generation of audiences that couldn't see themselves in the trials and persona of Hamlet. While it is certain that (most) people aren't visited by a ghost and told to avenge their murdered father's killer, everyone has had to face a task he or she didn't really want to do and didn't know how to accomplish. That is the crux of the story.
If you were to try and define a key adjective or two to describe Hamlet, you would be hard-pressed to limit yourself to just two AND you would find that if you really started brainstorming you would come up with 20 adjectives and some of them would contradict with others, and yet, they would all be true. That is the essence of human nature -- not simple definition but rich complexity. For example, Hamlet is usually considered contemplative as seen in his hesitation to act rashly, but he is also seen to act without thought of the consequences, such as when he slays Polonius. He is both witty and morose. He is both unschooled and old before his time. Like all humans, he is concerned with personal success, life, what happens at death or after death, and how to find a nice balance between taking the actions that one can while accepting the things of life that are merely one's fate. That is what makes Hamlet so real and so relevant to every generation.