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I'd disagree. I know it's often interpreted along those lines, but it seems a misread on the situation to me. There are quite a few literary critiques on Hamlet that view him and his dilemma as existential angst -- a hero torn between 'duty and doubt'. I think that reading is far more apt than viewing Hamlet as a suicidal emo fop. The very next lines after the famous intro are literally:

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

So... Whether it's nobler to sit there and get fucked by 'chance', or to take up arms and oppose the status quo/issue. That opening clearly establishes the two sides of his deliberations: to suffer the situation, or to take up arms and opposed it -- neither of these is equated to dying until the next part, which pivots to death, because he was opposing the king / considering killing his usurper uncle. Not only would that potentially result in his death, but the act of killing in the 'christian' mindset would result in his soul being damned in the next life. He spends a huge amount of the play humming and hawing about this sort of stuff, like when he has an opportunity to top the guy, but stops because his uncle is mid prayer -- and he doesn't want to kill him in a way that might accidentally send him on to heaven, if such a thing existed.

Anyhow, the next part supports my read, I think, where he goes through a list of "mundane" offenses. Thees offenses are basically all sleights that someone would be 'suffering' as a result of actions of another - they have an external locus, and there's no explicit reason to think that the 'response' with a bare bodkin (dagger) would be directed internally: the oppressor's wrong (tyrants), the proud man's boasting (we hear alot of boasting from certain folks...who are blind to the impact on others), the pain of being shunned romantically, the slowness of the law to achieve justice, the insolence of office (putting up with an idiot in a position of power), and the general pain of generally having to put up with those 'unworthy' of your efforts. His bridging line there is to finish the list with a note that you could fix most of those situations with a dagger, before finishing it off with:

Who would fardels bear, To grunt and sweat under a weary life, But that the dread of something after death,

That's pretty explicitly saying "why would anyone 'suffer these slings and arrows' (mundane offenses) if it weren't for a fear of death by 'taking up arms against them' with a dagger?" (to reference it back to the earlier start for cohesion in the reading, which works just fine).

In the speech he also equates inaction to cowardice, and that to effectively being dead. Near the end:

And makes us rather bear those ills we have Than fly to others that we know not of? Thus conscience doth 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.

Ie our fear of dying, the uncertainty of what's next, makes us bear those problems / put up with a lot of crap -- it makes us into cowards. Our resolve is diminished by the thought of the challenge / potential death, and the possibility of going to hell if you murder the source of these troubles, to the point that we lose the ability to take action.

And again, a huge amount of the play is literally all about Hamlet, trying to sort out the morality of whether he should kill his usurper father -- an act which he knows would put his own life in jeopardy and cause potential chaos - let alone put his own 'immortal soul' in jeopardy of going to hell, if he accepts the idea of heaven/hell. Not so much Hamlet debating if he should kill himself, but rather if he should kill his uncle. He's out for revenge, he's not out to be an emo baby.