How was the character Hamlet or any other characters, relatable to the younger generations of the Elizabethan era? Basically in what ways did the...

There are several characters in Shakespeare's Hamlet that the author uses to target the interests of young people during the Elizabethan (English Renaissance) period.

The first character is Hamlet. While Hamlet's father has just died and Hamlet is the Prince of Denmark, the throne has passed, instead, to his uncle Claudius. At that time, this would have been an acceptable procedure if Hamlet was believed to be too young or too experienced. Hamlet has been away studying at the university, and Claudius has obviously been judged to be better suited (than Hamlet) to assume the throne upon his brother's sudden death. Claudius assures Hamlet that he is still next in line to the throne:

...for let the world take note
You are the most immediate to our throne,
And with no less nobility of love
Than that which dearest father bears his son
Do I impart toward you. (I.ii.111-115)

This would certainly be something with which young people in the audience, specifically young men of nobility (though the audience was made up of people from all classes) could identify and empathize.

Another thing that would resonate with a young Elizabethan crowd would be the sense that one was expected to bow to the wishes of a parent, and also a king, pushing his or her own wishes aside.

Claudius is quick to tell Hamlet that he should put mourning behind him, which Hamlet (of course) does not appreciate. After all, Claudius sits where Old Hamlet once reigned, and more distressingly, has married Old Hamlet's widow—and Hamlet's mother—Gertrude. However, Claudius also does not want Hamlet to return to school, but to stay on in Denmark.

For your intent
In going back to school in Wittenberg,
It is most retrograde to our desire;
And we beseech you, bend you to remain
Here... (115-119)

We also see this pattern of socially accepted behavior as Laertes requests permission to be excused in order to return to France. As is appropriate, the King asks if Laertes has requested his father's permission (59).

Learning that this had been done, Claudius grants Laertes leave to depart from the royal court and Denmark:

KING:

Take thy fair hour, Laertes. Time be thine,
And thy best graces spend it at thy will! (64-65)

It is interesting to note that Laertes is allowed to leave to return to France—and he simply goes to have a good time! However, Hamlet, who desires to return to school is not allowed, and this has to irritate him as he does not like his uncle—in fact, he very much resents him. After all, it is obvious to Claudius and Gertrude that Hamlet is extremely unhappy. In that Claudius has asked Hamlet to look upon him as a father, we can infer that the King knows that Hamlet's dark mood has a great deal to do with his uncle. Keeping Hamlet at home might raise questions in the audience's mind: when we discover that Claudius is a murderer, might the sovereign not want Hamlet neatly out of the way at school so he is not there to question the circumstances surrounding Old Hamlet's death?

Ophelia, Polonius' daughter and Hamlet's sweetheart, is also at the mercy of her superiors, "." She is an obedient daughter who would not resist anything her father—or her king—might ask of her. We see Ophelia as a pawn of the men in her life. (In that era, women were not valued—considered weaker and less intelligent than men.) Polonius is not above using his daughter to spy on Hamlet, most importantly because he wants to impress Claudius. So with the King, Polonius arranges that they will hide behind a curtain ("arras") and eavesdrop on Ophelia and Hamlet as they speak.

I'll loose my daughter to him.
Be you and I behind an arras then;
Mark the encounter. (174-176)

Essentially, Ophelia is forced to spy on the man she loves. The idea of stepping away from one's personal desires is something a young audience of the time could identify with, most especially if they had been asked to turn their back on someone they loved so that it might benefit someone else—particularly from a financial or political standpoint. Ophelia's powerlessness would ring true to many of the young women in the audience, though most would know better than to complain, as they were at the mercy of the men in their lives regardless of their social standing.

These are things with which a young person from the Renaissance-era (or today) would not be pleased, and could well appreciate.