How does the play Hamlet treat the idea of suicide morally, religiously, and aesthetically? Why does Hamlet believe that, although capable of...

Hamlet grapples with the ideas of suicide and mortality quite a bit throughout the play. In his first soliloquy ("O but that this too, too solid flesh would melt") Hamlet is aware of the consequences of suicide from a religious stand point. The first lines do not point to a desire to directly kill himself, but rather a desire to simply perish - "O that this too too solid flesh would melt, Thaw, and resolve itself into a dew!" The following lines begin with the word "or." If Hamlet cannot spontaneously and simply stop existing, he wishes that suicide was not considered a mortal sin. This opening showcases a quality we see in Hamlet throughout the play - his inability to act. He spends a lot of time wishing things were different. The easiest thing would be to just stop existing through no action of his own. But he can't end his own life because he does not want to suffer the eternal consequences. So with these two options out, he has no choice but to continue living and being miserable.

In Act 3, scene 1, we see the same conflict in his "To be or not to be" soliloquy. He considers whether it is better to live and suffer or to commit to death and escape the suffering of life, but also face the unknown after death. He likens death to a sleep and the afterlife to dreams, saying,

"For in that sleep of death what dreams may come
When we have shuffled off this mortal coil,
Must give us pause."

He continues asking who would willingly go through a life full of suffering if they could find peace in death, unless there was something to be feared in the space after death. Hamlet clearly fears that he will suffer far more after he is dead than he is suffering in life. However, he is angry at himself for his "cowardice" and concludes the speech by saying, 

"Thus conscience does make cowards of us all,
And thus the native hue of resolution
Is sicklied o'er with the pale cast of thought,
And enterprises of great pith and moment
With this regard their currents turn awry
And lose the name of action." 

It is interesting that he refers to what stops a person from killing themselves as conscience. This points to a moral dilemma rather than a fear-based conflict. It indicates that on some level, Hamlet believes it is wrong to take one's own life and not just the fact that he is afraid of facing the eternal unknown. 

Finally, the play deals with the suicide of Ophelia. Act V, scene i opens with the clowns, or the gravediggers, commenting on the circumstances:

First Clown: Is she to be buried in Christian burial when she wilfully seeks her own salvation

Second Clown: I tell thee she is; therefore make her grave straight. The crowner hath sate on her, and finds it Christian burial.

First Clown: How can that be, unless she drown'd herself in her own defence?

Second Clown: Why, 'tis found so.

The gravediggers--simple laborers--have an awareness that according to the laws of Christianity, Ophelia should not be getting a Christian burial. The scene continues with the clowns further commenting that were it not for Ophelia's social status, she would not be getting a Christian burial. As Shakespeare's clowns often do, these characters present the "everyman's perspective" on Ophelia's suicide. 

Further into the scene, we see an exchange between Laertes and the priest conducting the funeral ceremony. Ophelia does not receive all the rights of Christian burial, and the priest indicates that it is more than enough that she is buried in holy ground. The characters that loved Ophelia do not openly acknowledge that her death was suicide at any point. Gertrude's speech says that Ophelia "fell" into the water, and the circumstances of her death are not spoken by any other character. This characterizes the death as taboo and unmentionable. Those that cared for Ophelia ignore it rather than acknowledge it.