William Shakespeare's Hamlet is a Renaissance hero (par excellence). Explain by making references to the play.

A Renaissance hero was one whose audience was Christian and thus had Christian morals and expectations of their heroes. They were not necessarily of noble birth, although Hamlet was, being prince of Denmark. Also, in comparison to classical tragic heroes, who had a fatal flaw and followed a rather linear path toward their death or comeuppance, Renaissance heroes were of more mixed moral stature and their demise was more complex. 

Hamlet himself is a Christian...

A Renaissance hero was one whose audience was Christian and thus had Christian morals and expectations of their heroes. They were not necessarily of noble birth, although Hamlet was, being prince of Denmark. Also, in comparison to classical tragic heroes, who had a fatal flaw and followed a rather linear path toward their death or comeuppance, Renaissance heroes were of more mixed moral stature and their demise was more complex. 

Hamlet himself is a Christian and helps create his own complex path by passing up the opportunity to kill Claudius while he's praying. He believes that if he kills Claudius while he is praying, Claudius will die absolved of his sins and thus Hamlet himself won't have the revenge he seeks, since the Ghost of his father has told him that he himself died in his sins and is thus: 

Doom'd for a certain term to walk the night,
And for the day confined to fast in fires,
Till the foul crimes done in my days of nature
Are burnt and purged away.

Hamlet wants the same punishment for his father's murderer (which is notably unChristian of him). He also is unsure of the ghost's accusations, and thus is determined to find out for himself whether Claudius did indeed murder his father while he slept--hence the play wherein the same act is performed, so Hamlet can watch Claudius' reaction. 

He cannot make up his mind what to do--his indecision is one of the main themes of the play--thus leaving room for a tangled plot. Hamlet finally decides he must kill Claudius, as the ghost has commanded, but in the process he accidentally kills Polonius (who was eavesdropping in the curtains), causing Laertes (his son) to seek revenge; his fiancee, Ophelia, goes mad and drowns; his mother drinks the poisoned wine he'd arranged for Claudius to drink and dies, and Hamlet finally stabs Claudius, and accidentally kills Laertes (because Laertes has poisoned his own sword, and the swords were swapped in the duel). Hamlet dies from the poisoned sword himself (and drinking what's left of the poisoned wine, to make sure). He leaves a path of bloodshed behind himself because he chose to not kill Claudius when the death would have been clean.