How does Hamlet fit into the Renaissance idea of man? Consider how his society is influenced by Renaissance ideals and how it affects his success.

The Renaissance is considered a bridge between the Middle Ages and modern history. It is characterized by the philosophy of humanism, by which man becomes primary. More radical ideas began developing in all aspects of life, such as art, science, religion, and politics. And in this respect, Hamlet is considered the perfect Renaissance man.
Hamlet expresses numerous humanist ideas, such as in act 2, Scene 2:

What a piece of work is a man! how noble in reason! how infinite in faculty! in form and moving how express and admirable! in action how like an angel! in apprehension how like a god! the beauty of the world! the paragon of animals!

Here, he praises the human mind and asserts his humanist ideas. His approach to the world is much different than that of medieval characters, and he continues to depart from traditional medieval religious ideas with his most famous soliloquy, which comes from act 3, scene 1:

To be, or not to be—that is the question:Whether 'tis nobler in the mind to sufferThe slings and arrows of outrageous fortuneOr to take arms against a sea of troubles,And by opposing end them. To die—to sleep—No more . . .But that the dread of something after death—The undiscover'd country, from whose bournNo traveller returns—puzzles the will . . .

Hamlet talks about the uncertainty of afterlife. He clearly questions the existence of Heaven and Hell, and is open to different interpretations of what happens after death. He questions life and its mysteriousness and mortality. Hamlet is not a risk-taker—which is why he refrains from taking his own life, as he does not know for certain what awaits a person after death. He calls everything into doubt, and this makes him the typical Renaissance philosopher.
Ophelia asserts that Hamlet's mind was once noble, but that now he has gone mad. She believes he had been a prince, heir, role model, soldier, scholar, courtier; he had been good as fencing, good at speaking—the perfect Renaissance man:

O, what a noble mind is here o'erthrown!The courtier's, soldier's, scholar's, eye, tongue, sword;The expectancy and rose of the fair state,The glass of fashion and the mould of form,The observed of all observers—quite, quite down! (act 3, scene 1)

Therefore, Hamlet is considered the iconic Renaissance man due to his critical, even radical outlook on life. His ideas express the changing values of the Renaissance era and give insight as to his humanist beliefs.