How did Gertrude hurt Hamlet?
By marrying Claudius, Gertrude helps to legitimize him as the lawful ruler of Denmark. Claudius managed to get the electors to choose him in place of Hamlet, who was the obvious heir apparent, by persuasion, bribery, intimidation, or whatever other means this clever villain used. But Gertrude hurt her son by marrying Claudius and helping him to look more legitimate by being married to the already reigning queen.
Claudius keeps assuring Hamlet that he is next in line of succession. This was one of the arguments Claudius used to get elected while Hamlet was away at Wittenberg. Claudius said that Hamlet would still become king after his death. But in Act 3, Scene 2 when Claudius asks, "How fares out cousin Hamlet?" Hamlet deliberately misconstrues the word "fares" to mean "eats" and replies:
Excellent, i' faith; of the chameleon's dish. I eat
the air, promise-cramm'd.
Hamlet is saying, in effect--although his uncle does not understand him and thinks he is mad--that he does not trust Claudius' promises to have him inherit the throne. Claudius may have promised Gertrude that her son would still become king in order to get her to marry him. Hamlet knows there are at least two ways in which he could be cheated out of his rightful inheritance. One would be if he fell out of favor with Claudius. Another would be if Gertrude bore Claudius a child. Hamlet hates the torrid love-making that goes on between his mother and his uncle. He not only considers it adulterous, but he knows intuitively that if his mother became pregnant--which still seems possible--then Claudius would give preference in succession to their child, regardless of whether it was a boy or a girl. In fact, Claudius might decide to have Hamlet murdered to forestall any controversy about succession after his death. If Claudius can murder Hamlet's father, then he can certainly murder Hamlet.
That is a second way in which Gertrude hurts Hamlet. The third way is related to that second way. She breaks Hamlet's heart by the way she forgets his father so quickly and engages in what her son considers adulterous, incestuous, and disgusting sexual orgies. In Act 3, Scene 4, he expresses his pent-up loathing for her behavior as well as his hatred for his uncle.
Nay, but to live
In the rank sweat of an enseamed bed,
Stew'd in corruption,honeying and making love
Over the nasty sty!
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A murderer and a villain!
A slave that is not twentieth part the tithe
Of your precedent lord; a vice of kings;
A cutpurse of the empire and the rule,
That from a shelf the precious diadem stole
And put it in his pocket!
His mother's fallen condition and her shameful betrayal of her deceased husband's memory are perhaps what hurt Hamlet most of all. He feels personally dishonored as her son and as the son of his noble father. He shares the feelings of the Ghost, who arrives on the scene shortly after he has spoken the above words to his weeping mother.
So Gertrude hurts Hamlet by marrying Claudius and helping to legitimize him as king. She hurts Hamlet publicly by urging him to forget all about his father and accept the new status quo. She hurts Hamlet by engaging in sexual intercourse with his hated uncle which could result in the conception of a child who would certainly destroy Hamlet's chances of ever becoming king. And she hurts her son most of all by dishonoring his father's memory in what he considers adultery, incest, and vile debauchery.