How does Hamlet treat the idea of suicide considering the “O, that this too too solid flesh would melt” soliloquy (I.ii.129–158) and the...

That Hamlet suffers greatly from depression is apparent.  His statement in Act I wishing that his "solid flesh would melt" is typical of severely depressed people.  For, melancholy causes its victim to become passive.  So, in their wishes to eschew their troubles, they think that it may be the best thing if they did not wake from sleep; thus, Nature would take the active part and remove the worry from them of what to do with their lives: 

To die, to sleep--/No more; and by a sleep to say we end/The heartache, and the thousand natural shocks/That flesh is heir to:  'tis a consummation /Devoutly to be wished.(60-64) 

However, there's a "rub" to the comfort of "sleep." If one induces it oneself, he is guilty of committing suicide, which is a sin, punishable in the life after death:

To die, to sleep--/To sleep, perchance to dream, ay there's the rub/For in that sleep of death what dreams may come/When we have shuffled off this mortal coil/Must give us pause.(64-67) 

As Hamlet's depression increases and he spends more and more anxious moments contemplating the revenge his father's ghost requests, Hamlet realizes that regicide is a very serious act, an act for which he can be punished by death as one who has committed treason.  Also, suicide condemns one to hell. This is why Hamlet "gives pause."  Hamlet reflects that in this pause, one tolerates

The oppressor's wrong, the proud man's contumely,/The pangs of despised love, the law's delay,/The insolence of office, and the purns/That patient merit of th'unworthy takes(III,i,71-74)

The dread of what lies beyond the grave deters the melancholic Hamlet. However, all his inaction and self-debate proves self-defeating to Hamlet as "conscience...make(s) a coward of [him]."

After the soliloquy in which Hamlet contemplates suicide, he is riddled by inaction and moments of irrationality and ridiculous humor.  Timothy Bright writes in his "Treatise of Melancholy":

Sometime it falleth out that melancholy men are found very witty, and quickly discern, either because the humor of melancholy with some like sort with their spirits...are instruments of such sharpness. To this, other reasons may be added,as : exercise of their wits, wherein they be indefatigable, which maketh them seem to have that of a natural readiness which custom of exercise, and use hath found in them.  Moreover,...melancholy breedeth a jealousy of doubt in that they take in delibertation...Their resolution riseth of long deliberation, because of doubt and distrust which...disturbeth the sleep of melancholy persons.

As the plot "Hamlet" progresses, the reader perceives this procrastination in Hamlet and continual debate with death.  Finally after talking with Fortinbras who is willing to sacrifice himself and his army for a cause, Hamlet, in his final soliloquy, resolves to act for a cause much stronger than that of Fortinbras and is, thus, shaken from his melancholy and thoughts of suicide:

How stand I then,/That have a father killed, a mother stained,Excitement of my reason and my blood,/And let all sleep, while to my shame I see/The imminent death of twenty thousand me/That for a fantasy and trick of fame/Go to their graves like beds....Oh, from this time forth,/My thoughts be bloody, or be nothing worth! (IV,iv,56-66)