How is "life-redeeming" seen in the character of Hamlet, in Shakespeare's Hamlet?
In terms of Shakespeare's Hamlet, we can see that Hamlet's choice to live his life on his terms and not on Cladius' terms, could be considered "life-redeeming."
"Redeem" means to "get or win back."
When Hamlet returns home from university studies for his father's funeral, he is greeted quickly by his mother's swift remarriage to his uncle (which was considered by Elizabethans to be incestuous, even though Gertrude and Claudius were only related previously by marriage). Gertrude is fawning all over Claudius, having exchanged her mourning for Old Hamlet for the flirtatous and happy actions of a new bride.
Why, she would hang on [my father]
As if increase of appetite had grown
By what it fed on; and yet, within a month…
A little month, or ere those shoes were old
With which she follow'd my poor father's body
Like Niobe, all tears—why she, even she—
O God! a beast that wants discourse of reason
Would have mourn'd longer—married with my uncle… (I.ii.146-149, 151-154)
His uncle insists that Hamlet stop grieving for his father: that to continue grieving is stubborn, showing "unmanly grief" (97). Hamlet silently refuses to do so. He cannot wrap his brain around what has happened to him. He feels as if his world has crumbled around him and madness has taken over: his mother is a stranger; his father is gone; and, his uncle is now his stepfather and King. It is not until scene five that Hamlet learns from what appears to be his father's ghost that Old Hamlet was not killed by the bite of a serpent, as is widely believed, but has been murdered by his own brother, Claudius, while he napped.
Revenge his foul and most unnatural murder… (I.v.29)
Murder most foul, as in the best it is;
But this most foul, strange, and unnatural. (31-32)
[…] know, thou noble youth,
The serpent that did sting thy father's life
Now wears his crown. (43-45)
Hamlet realizes that the Ghost could be a spirit of evil trying to draw him to his eternal damnation by killing Claudius: for to kill a king is a mortal sin. So Hamlet goes about finding a way to prove his uncle's guilt. He uses players that visit the castle to reinact his father's murder, watching to see how Claudius will react—and Claudius' guilt is quickly made evident when he sees his actions portrayed on stage.
It is in taking steps to expose his uncle that Hamlet grows up: he sees the seedy underbelly of life, which he had never (we can infer) seen as son to Old Hamlet and Gertrude. Claudius introduces the evil in the story, by killing his brother, who is also the King of Denmark. Hamlet's decision to avenge his father's death is "life-redeeming," for he takes back what is left of the life he had before Old Hamlet died, choosing to risk all: his love for Ophelia, his affection for his mother (albeit, badly damaged) and his life. In his actions, Hamlet chooses to lead life on his terms and not based upon the conditions created by his uncle. Had he done so, Claudius would have "won," becoming King and getting away with murdering his own sibling.