This line from Shakespeare's Hamlet, is one of the most powerful single lines, I think, in the play.
By the time Laertes makes this statement about killing Hamlet, Old Hamlet has been murdered and things are "rotten in Denmark." It is safe to say that this royal family is as dysfunctional as they come. In the midst of the problems of the royal family come the death of Polonius and Ophelia's insanity. It is easy to understand how upset Laertes would be, and how cleverly Claudius places all the blame on Hamlet, in the hopes that Laertes will do his dirty work for him. Laertes isn't such a bad person: he is devastated by the loss of his entire family. (He comes around later, when it's too late.)
Claudius carefully lays the groundwork, giving Laertes all the reasons why Hamlet is to blame, and why Claudius can't do anything about it himself. In watching Claudius "spin his web" around Laertes, we can see how Hamlet is ultimately outwitted by the King and killed—for Claudius has more experience in murder and deceit than Hamlet does—Claudius is an evil man; Hamlet is no match for the villainous Claudius.
We get a sense of Laertes' frame of mind in the following passage, where he describes what warms his heart:
But let him come.
It warms the very sickness in my heart
That I shall live and tell him to his teeth,
'Thus diddest thou.'
Claudius does his work well, capitalizing on Laertes' grief and righteous need for revenge. Claudius then puts the question to Laertes as to how far he is willing to go to avenge Polonius' death—or is it all just talk on Laertes' part?
What would you undertake (135)
To show yourself your father's son in deed
More than in words?
This speaks to the character of Laertes, in response to this question— where Laertes admits that he would face any kind of censure to be satisfied, with the death of Hamlet. Laertes is stating that he would risk God's anger to kill Hamlet—is not just a lot of words, as Claudius insinuates.
To cut his throat i' the church.
There is a great deal of historical significance in this quote. There is a long-standing belief that a church offers safety and protection to those who pass through its doors.
"Sanctuary" was also a right to be safe from arrest in the sanctuary of a church or temple, recognized by English law from the fourth to the seventeenth century.
As an example, when Henry II's men killed Thomas Becket, the Church's Archbishop of Canterbury [Cathedral], besides his death which was appalling was the fact that he was killed in the church, a place of peace, refuge and safety.
The frightened monks persuaded Thomas to flee from his residence towards the Cathedral where they felt that he would be safe.
So when Laertes claims he would kill Hamlet in a church, he is defying the laws of the church to consider such a thing. Claudius says a church is not a place for murder—that if killing were allowed in a church, there would be no end to murder, intimating that murder is prohibited in a church.
No place indeed should murder sanctuarize;
Revenge should have no bounds.
Laertes' response also speaks to the diabolically effective job Claudius has done to stir Laertes' need for vengeance, that he would kill Hamlet in a holy place, so great is his desire to see Hamlet dead.