In "Hamlet" broadly describe how the conduct of Gertrude annoys Hamlet and makes him sad?
Sad is a bit of an understatement, to say the least. Hamlet expresses more rage, disgust, repugnance and bitterness at his mother's actions than sadness. At the very beginning of the play, he moans and groans about how Gertrude used to
"hang on [Hamlet's father] as if increase of appetite had grown by what it fed on: and yet, within a month...married with my uncle." (I.ii.143-151)
So, he is disgusted by his mother's quick marriage to his uncle after his father's death. He is confused, because she seemed so in love with his father, but only 4 weeks after he dies, she remarries. He calls this action "wicked", "incestuous", "unrighteous", and frail (I.ii.154-158). He is so angered by it that he tries to tell himself, "Let me not think on't" (I.ii.146) because it is so infuriating to him. Later, after the ghost visits him, he calls his mother a "most pernicious woman" (I.v.105), venting his anger once more. In her bedchambers after the play, he unleashes another rant, this time to her face, saying that what she had done "blurs the grace and blush of modesty" (III.iii.41) and goes on and on about how awful it is that she married so quickly to his uncle.
In more broad terms, Hamlet is upset because her marrying so quickly makes it seem like she really didn't love his father; it is a disrespect to the great man that he was. It is her basically saying, "Eh. He was nice, but, not worth mourning over." He is insulted for his father. Then, for her to marry his father's brother, he is even more disgusted, because to him, they were related. He calls it incest, even though it isn't technically correct. He just finds Claudius to be inferior in every way compared to his father, and for Gertrude to marry him is just insulting and gross. By all accounts, she seems completely happy in her new role as wife to Claudius; she teams up with him, urging Hamlet to stop mourning. She is happy at feasts and in public gatherings. All of this rankles Hamlet, and just adds fuel to the fire of his anger.