Why is Hamlet unhappy?

Shakespeare’s Hamlet unfolds with a melancholy environment, as the young Hamlet enters the stage troubled by the recent death of his father and marriage of his mother to his uncle Claudius.  Hamlet is greatly perturbed by the events as conveyed in his first exchange with Claudius:

Claudius: But now, my cousin Hamlet, and my son—
Hamlet: (aside) A little more than kin and less than kind.
Claudius: How is it that the clouds still hang on you?
Hamlet: Not so, my lord.  I am too much i’ the sun. (I. ii. 64-67)

In this exchange, Claudius refers to Hamlet as both cousin and son, which angers Hamlet. Hamlet says that Claudius is now “more than kin,” as he is now his stepfather, but quickly adds that Claudius is “less than kind.”  This is in part due to the fact that Claudius has so hastily married Gertrude, as Hamlet points out, “The funeral baked meats / Did coldly furnish forth the marriage tables” (I. ii. 179-180). 

Indeed, the death of his father and remarriage of his mother have a great effect on Hamlet, and he spends the majority of the play consumed with grief.  He explains:

'Tis not alone my inky cloak, good mother,
Nor customary suits of solemn black,
Nor windy suspiration of forced breath,
No, nor the fruitful river in the eye,
Nor the dejected 'havior of the visage,
Together with all forms, moods, shapes of grief,
That can denote me truly. These indeed “seem,”
For they are actions that a man might play.
But I have that within which passeth show,
These but the trappings and the suits of woe. (I. ii. 77-86)

In this passage, Hamlet explains that his mourning, dark clothes, weeping and sighs are just surface expressions of the true grief he feels.  He is truly angered and upset about his father’s death and mother’s hasty remarriage. He goes on to state, “Within a month, / Ere yet the salt of most unrighteous tears / Had left the flushing her galled eyes, / She married” (I. ii. 154-57).  In this hyperbole, Hamlet claims that his mother remarried before her tears over her husband’s death even dried.  He then states, “It is not nor it cannot come to good, / But break, my heart, for I must hold my tongue” (I. ii. 159-60).  Thus, Hamlet is unhappy because not only has his father died and his mother married his uncle, he cannot adequately express his melancholy or objection.  He therefore spends his time wandering around the castle in mourning clothes, contemplating the death of his great father and loathing the fact that his mother has so readily remarried.