Hamlet's mother, while his father was alive, was completely enamored with her husband the king. Hamlet himself describes how "she would hang on him, as if increase of appetite had grown by what it fed on" (I.ii.143-144). So after her husband dies, the "expected" reaction would be devastated grieving for a very long time. If she was so in love with him, his death would have shattered her. However, Hamlet is very upset because "yet, within a month-Let me not think on't-Frailty, thy name is woman!-A little month...married with my uncle, My father's brother...within a month...she married" (I.ii.145-155). He is disgusted because she, whose love was so strong for his father, with seeming no regard or mourning, quickly married his uncle. First of all, it is an insult to his father, and secondly, it is his father's uncle, which Hamlet considers "incestuous" and vile.
He becomes even more disgusted when the ghost informs him that it was most likely his uncle that killed his father. So not only did Gertrude marry his father's brother, but a man who was a murderer to boot. His anger towards his mother flavors much of the mood of the play; because of it, he unleashes an angry tirade upon Ophelia about the fickle nature of all women, which contributes in part to her unfortunate turn to madness, and later he verbally assaults his mother after the play.
Hamlet is disgusted with his mother's rash act of remarrying so soon after his father's death. He tells her "Mother, you have my father much offended (II,iv,9). She seemed to love him, yet she supposedly fell in love with his brother? Perturbed at this rash, almost incestuous act, Hamlet says to her,
Here is your husband, like a mildewed ear/Blasting his wholesome brother. Have you eyes?....You cannot call it love, for at your age/ The heyday in the blood is tame, it's humble,/ And waits upon the judgment, and what judgment/ Would step from this to this/...What devil was't/That thus hath cozened you at hoodman-blind?...O shame, where is thy blush? (III,iv,68-82)
Here he also tells his mother she is too old to have lustful feelings and should be ashamed to not use reason in her judgment.
In addition to what has been observed above, it is important to understand that in Shakespeare's England, such a marriage as that between Gertrude and Claudius was a breach of canon law (the law legislated, interpreted and enforced by the Church, in this case the Established [or Anglican] Church) and was, in fact, considered incest. It is not that it seemed incestuous. According to canon law, it was incestuous. So, even though the setting is Denmark, Shakespeare's audience could clearly understand Hamlet's disgust with such a relationship.