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.