I the play Hamlet, how do Hamlet's soliloquies and asides give us insight into his melancholic character, relating to the idea of Revenge Tragedy?
Shakespeare uses soliloquies and asides so that the audience may hear what the character is thinking, while the other characters on the stage do not.
In general, Hamlet's thoughts lean mostly toward his sorrow for his father's passing, his distrust of his uncle and thoughts of how to prove him implicit in Hamlet's father's death, his lost appreciation for the beauties of the world around him, and even his consideration of suicide.
For example, Hamlet discusses how the world "delights not me," having lost his father and seen his mother marry her brother-in-law so soon after her husband's death. His depression has made him cynical.
Later, Hamlet privately admits that he will put on "an antic disposition," pretending to be driven mad by his father's death. In doing so, he hopes that Claudius will slip up and give something away. Hamlet plays word games with Polonius who is too foolish to understand what Hamlet is alluding to, but Hamlet sees much more meaningful results when he has the players reenact the murder of his father when the acting troupe performs for the new king.
After this, Hamlet finds his uncle at prayer and is ready to kill him, however he mistakenly believes that Claudius is confessing his sins and fears that if Hamlet kills him at that moment, Claudius, with a "clean" soul, will go to heaven. (In truth, Claudius cannot find the words to pray, but Hamlet doesn't know this.) Hamlet vows that in order to get revenge, he must kill Claudius after he has done something sinful...like sleeping with Gertrude, Hamlet's mother. The Elizabethans of Shakespeare's time believed that when Claudius married his sister-in-law, he committed an act of incest. Therefore, Hamlet plans to wait until Claudius has been to bed with Gertrude and then kill him.
Hamlet suffers from extreme depression (melancholy). In his "To be or not to be" speech, he contemplates suicide. His love for Ophelia falls to the wayside. He is distanced from his mother, unsure until his father's ghost tells him, whether his mother was complicit as well in the death of Old Hamlet.
His pretended madness, his mistaken murder of Polonius, and Ophelia's subsequent suicide are all manifestations of Hamlet's desire for revenge that overshadows all other aspects of his life. Indeed his cynicism for the world and his denial of Ophelia are examples of his intent to find justification--and the ideal time--to kill the king to avenge his father's murder.
His soliloquies and asides express his melancholic nature. He commits himself to his father's charge above all else (wipes out all other books in his mind); he distrusts others (that one can smile and smile and be a villain); he berates himself with guilt (how can the players act their part so well and I can't--I'm a horrible son), he has a hard time starting a task (the play's the thing, wherein I catch the conscience of the king), he analyzes philosophically (to be or not to be), and he seeks perfection (won't kill Claudius until he's sure of sending him to hell).
All of these traits enhance the Revenge tragedy, for he is committed to revenge, his delays create suspense but never doubt in his purpose, his distrust prompts vivid descriptions of gore, he sees himself as a victim, isolates himself, and his inward deliberations increase his passion--he grows bloodthirsty (now could I drink hot blood). His intelligence lends to tricks of language and sick humor.
That's all I can think of off the top of my head. Hope this helps. You'll probably want to clarify my paraphrasings with the exact quotations.