What is Hamlet's mood or attitude in his first appearance in the play? What is bothering him?

Hamlet is deeply unhappy and depressed when we first encounter him act 1, scene 2. Claudius, in fact, asks him why "clouds still hang" on him and why he is still wearing black. Claudius tells him that the death of his father is part of the rhythm of life and says that he should get over it. His mother says the same, asking why he still seems sad and mournful.
Hamlet's attitude is one of frustration as he bursts out that he still "seems" unhappy because he truly is unhappy. Other people may be pretending to be sad and putting on the "suits [outer appearance] of woe," but for him the grief is real. It is more profound than what he lets appear on the outside. Hamlet states,

I have that within which passeth show.

After he is left alone, Hamlet delivers his first soliloquy, in which he states that he wishes he could kill himself. He expresses the deep depression he feels by saying,

How weary, stale, flat, and unprofitable
Seem to me all the uses of this world!

He further explains that he does not respect his uncle or think his uncle is half the man his father was. He is distressed that Claudius is now king of Denmark and very upset that his mother not only married Claudius, but married him so soon after his father's death, stating,

She married. O most wicked speed, to post
With such dexterity to incestuous sheets!

Hamlet's soliloquy shows that he was not exaggerating when he told Claudius and his mother that they didn't begin to understand his misery. He is far more upset and depressed than they recognize. The whole world seems "rank," or rotten, to him, and he can't get past his grief at losing his father or his fear that his mother, in remarrying so quickly, might not really have loved his father. His whole world has been turned upside down.

It's important to note that while we as an audience have seen the ghost of his father, Hamlet has no idea of this until later in the scene when Horatio tells him about it. His sadness now is wholly about his father's death and his mother's remarriage.