Ultimately, Hamlet speaks rudely to these four characters in Act III. At the beginning of Act III, Hamlet gives his famous soliloquy where whe contemplates suicide and ponders whether it is better to live a life of sorrow and mental (and potentially physical) torment or to die and face whatever comes after death, the "undiscovered country"(3.1.79). Ophelia comes upon him, and he speacks rudely to her, telling her he does not love her and never did love her (3.1.114, 117). During the play within the play, he speaks rudely to Ophelia again, making sexual puns in lines 100-114. He also treats his mother rudely by not sitting next to her and by accusing her of offending his father (3.4.9) and blaspheming her behavior throughout his speech to her in Act III, Scene iv.
When Hamlet speaks to Polonius, he treats him dismissively and as a fool, simply prattling to him about animal shapes clouds can make (3.2.349), and once he kills Polonius, he does actually call him a fool (3.4.33). With Claudius, Hamlet speaks nonsense to him before the play within the play and then speaks with hatred and a desire to kill him when he sees Claudius praying. However, Hamlet decides not to kill Claudius at this moment because he worries that if he kills him while he's praying, Claudius will go to Heaven, and Hamlet does not want that to happen (3.3.73-96).
By this time in the play Hamlet is putting his plan into action to make Claudius pay for killing his father the king. In act 3 Hamlet is behaving in a rude and confrontational manner. He is condescending and insolent to everyone. At the beginning of Act 3 he is gentle with Ophelia, but then becomes angry and vulgar with her later in the act. (see example below) He is sarcastic with his mother and Claudius. Hamlet also uses a great deal of word play with Polonius. Then at the end of Act three Hamlet kills Polonius who is hiding behind a tapestry and accuses his mother of helping to murder the King.
Do you think I meant country matters?
I think nothing, my lord.
That's a fair thought to lie between maids' legs.
What is, my lord?
You are merry, my lord.
Ay, my lord.
O God, your only jig-maker. What should a man do
but be merry? for, look you, how cheerfully my
mother looks, and my father died within these two hours."