In Hamlet, how do Claudius and Polonius define their ideas about honour in this play?
Well, it's not a concept that ever gets very clear definition in "Hamlet". But you can figure a little bit out. Honour seems to have something to do with seeming noble and honourable, as Polonius tells Ophelia, when he is telling her not to respond to Hamlet's romantic advances:
I must tell you
You do not understand yourself so clearly
As it behooves my daughter and your honour.
What is between you?
It's not enough, he says, for Ophelia to understand what she wants, but she must understand what belongs to her reputation as his daughter, and someone honourable. Moreover, Polonius says, she must
...weigh what loss your honour may sustain
If with too credent ear you list his songs,
Or lose your heart, or your chaste treasure open
To his unmast'red importunity.
Dishonour is a stained reputation. Dishonour is people not thinking of you as squeaky clean. And Polonius, the master of keeping up these appearances, but spying on peopel in secret, hidden behind walls and behind curtains, of course, is the person Claudius thinks most honourable:
Polonius. What do you think of me?
Claudius. As of a man faithful and honourable.
Claudius can smile and smile and still be a villain. Appearances are deceptive. And so is "honour", at least, "honour" as these two define it.