The opening of the speech certainly gives a strong sense of desire for death:
O, that this too too solid flesh would melt
Thaw and resolve itself into a dew.
Or that the Everlasting had not fix'd
His canon 'gainst self-slaughter!
This words are a simple wish to either die on the spot, "to melt," or to be allowed by the tenants of his faith to commit suicide. He goes on to explain that this desire is because he finds the world to be "an unweeded garden," and that he has no use for such a world.
The other theme regarding death that Hamlet focuses on in this soliloquy is how quickly his dead father was forgotten by his mother. He mentions it at least twice:
But two months dead. Nay, not so much, not two...
...and yet, within a month
...A little month.
This focus on the time between his mother's following his dead father's body in the funeral and marrying his Uncle is a fact that he cannot forget. It is very important to Hamlet, in his concept of death, that the dead have their time to be remembered in a period of mourning. He goes on to comment on her haste:
O, God! A beast that wants discourse of reason
Would have mourned longer...
And his final conclusion about this hasty marriage is that:
It is not, nor it cannot come to good.
So, though Hamlet begins the soliloquy wishing for his own death and contemplating suicide, it becomes clear that his real concern is the lack of respect shown to his dead father by his mother, an illustration of his feelings about the importance of proper mourning over the dead.
One measure of Hamlet's growth through the play is his how Hamlet slowly comes to a full understanding of death. Death for Hamlet moves from the abstract to grim reality over the course of the play. The opening lines of this soliloquy show not only his naivete but reinforce the symbolism of his feelings. Of course the human body does not melt, thaw and resolve itself into a dew. We turn to dust. There is a marvelous little discussion in the graveyard after the Yorick speech where Hamlet realizes "to what base uses we may return." The noble dust of Alexander stops up a beer barrel, Julius Caesar turned to clay might stop a hole to keep the wind away.
This soliloquy is often misinterpreted as suicidal ideation but it is just youthful hyperbole. Because the part of Hamlet is always played by middle aged men, it lends the impression that this soliloquy is a product of a fully mature mind. The text clearly favors the contrary.