Hamlet sums up his requests of his mother in the following lines:
Confess yourself to heaven;
Repent what's past; avoid what is to come;
And do not spread compost on the weeds
To make them ranker.
What he wants of his mother is for her to admit that marrying Claudius was a sinful and awful decision. It was an insult to King Hamlet's memory and needs to be atoned for. He next wants her to repent for that bad behavior -- through prayers and actions. He lastly wants her avoid spending any more time and effort on Claudius. He gives her specific advice about avoiding her sleeping with him, saying that each abstinence will make the next easier.
A final request is that she not reveal the truth that Hamlet is only acting mad, which he reveals at the end of the scene.
His criticisms are mainly along the lines of questioning, how is it that she could go from this "hyperion" to a "satyr" to quote his earlier lines. He shows her a picture of his father and then a picture of Claudius and discusses how the two are so different and wonders again what was going through her head.
His requests to her are as follows:
To confess and repent of her sins so that she might be forgiven. He also tells her that she ought not to sleep with Claudius any more, she can make up an excuse if she likes but don't give in to his lust. He also tells her to make sure that she doesn't tell Claudius that he is actually sane, keep him thinking that he is nuts.
Hamlet is raging that his mother, a recent widow, has so quickly married his father's killer and is behaving like a horny teenager with her embarrassingly quick new husband.
He tells her to stop sleeping with Claudius and to remember her 'real' dead husband and to show him respect.