In Hamlet, why doesn't Hamlet kill Claudius in Act III when he has the chance?

You are of course
refering to Act III scene 3. This is whenis stricken by his guilt after seeing The Mousetrap
play especially put on for him by , and goes to pray to God.passes him by on his way to his
mother's chamber and has the perfect opportunity to kill him. Note what he says to himself as he
surveys the praying Claudius:

Now might I do it pat, now
he is praying,

And now I'll do't, and so he goes to Heaven,

And so am I reveng'd: that would be scann'd,

A villain kills my
father, and for that

I his sole son, do this same villain send

To Heaven.

The belief in Elizabethan times was
that if you were killed whilst praying you would go straight to heaven and bypass the fires of
Purgatory, where your sins during life were dealt with. Hamlet therefore chooses not to kill
Claudius because he wants Claudius to suffer for his sins and actions during his life, rather
than merely kill him and give him a ticket straight to heaven. He does not want to kill Claudius
when "he is fit and season'd for his passage" as he says in the same scene. That would
be no revenge whatsoever, especially as his own father's ghost is still languishing in purgatory
as he tells Hamlet.