Can you give me 2 points and proofs about how suicide is an important theme in Shakespeare's Hamlet?Can you discuss how the play treats the idea...

Suicide is an incredibly important theme in Hamlet, especially in regards to the character of Ophelia in addition to the main character of the play. 

Although the title of the play does not bear her name, Ophelia's importance cannot be denied when speaking of the aesthetics of suicide within Hamlet.  Scholars go back and forth about whether Hamlet's true flaw was inaction, but the irony here is that Hamlet contemplates suicide while Ophelia actually commits suicide, proving her to be the stronger character.  The key to Ophelia's suicidal beauty can be found within Gertrude's speech at the funeral:

Therewith fantastic garlands did she make / Of crowflowers, nettles, daisies, and long purples, / . . . / When down her weedy trophies and herself / Fell in the weeping brook.  Her clothes spread wide, / And mermaidlike awhile they bore her up / . . . / Till that her garments, heavy with their drink, / Pulled the poor wretch from her melodious lay / To muddy death. (4.7.168-183)

Of course, this scene is incredibly well done in may versions of Hamlet, my favorite being the Mel Gibson version.  In any case, there is no doubt of the beauty surrounding Ophelia's death, . . . as well as the beauty that surrounds her tenacity in this dreadful endeavor.

However, one cannot speak of suicide within Hamlet without mentioning the most famous speech in the English language.  Therefore, one must approach the character of Hamlet now and explore the moral and religious aspect of suicide from Hamlet's point of view.  Hamlet tells us he is contemplating suicide within the first line of the speech:  "To be, or not to be: that is the question" (3.1.56).  However, the key to the moral quandary can be found within the next few lines:

Whether 'tis nobler in the mind to suffer / The slings and arrows of outrageous fortune, / Or to take arms against a sea of troubles, / And by opposing end them. (3.1.57-60)

To approach the moral aspect of the question, Hamlet muses about what is "noble."  How interesting.  Be fortune's fool and live out your life in misery or win the fight by killing yourself.  How very full of optimism.  I have always felt that Hamlet's moral decision was that it was more noble to commit suicide (thus the irony in regards to Ophelia's death), but the point can be argued either way. 

Now, one must admit that there are quite a few of Hamlet's rants in this speech that we can skip.  However, it isn't long before Hamlet approaches the religious aspect of death:

To sleep--perchance to dream: ay, there's the rub, / For in that sleep of death what dreams maycome / When we have shuffled off this mortal coil, / Must give us pause.  (3.1.65-68)

An entire movie (based on what might happen to a soul after suicide) was based on four simple words:  "what dreams may come."  Hamlet approaches the afterlife here, although he does not mention it as heaven or hell per se.  Still, the thought of hell is enough to make him "pause."  Hamlet confirms his worry when he speaks of the following:

The dread of something after death / The undiscovered country, from whose bourn / No traveler returns. (3.1.78-80)

Suicide (or more appropriately the "dream" that comes after suicide) is simply fear of the unknown.  It is this fear that makes Hamlet stop short of ending his own life, . . . as Ophelia metaphorically completes his thought.