One of the best ways to show that a character has crossed the Despair Event Horizon is to give him/her the occasion to perform a Despair Speech to show that his/her hopes are lost and that there's nothing else left to do. Expect Creepy Monotone if the character is portrayed as scary.
A Despair Speech would be done by a character to show that they have crossed the Despair Event Horizon. The character is winding up his/her own lack of chances, desperation and, sometimes, remorse for something he/she could have done or said before. If said character is a criminal, his/her confession can become a Despair Speech. Be careful when writing this: done too much or badly, and it becomes Wangst.
Contrast Badass Boast.
Lelouch from Code Geass near the end of Turn 20: "My name is Lelouch vi Britannia, I am the eldest son of Empress Marianne, the prince who was abandoned by his empire. If anyone wishes to stop me, let them try, if there is anyone who can go beyond my despair."
Near the end of Episode 12, Marry delivers one of these after the Wide-Open Eyes Snake possesses Konoha and nearly kills Kido, Kano and Seto, and she resolves to reset time to prevent their deaths. She doesn't go through with it.
I've reinvented suicide as a group activity. Go me.
Batman, of all people, gets a borderline one in Going Sane as he tries to tell his girlfriend that he's finally realized that the world doesn't actually need Batman and he can let that part of his life go (without actually admitting he was Batman). Subverted when she not only reveals she knew he was Batman, but the world absolutely needs him so he better get his ass in gear and go back to Gotham.
Hal Jordan got a few of these back during his Parallax days.
Bilko: Well, of course I have a plan! [spelling out PALN on the chalkboard behind him] a P-L-A-N plan!... but maybe, a plan is not what I really need [gets down on his knees], what I really need is just a little puppy. A little puppy with big brown eyes, who would just come to me and lick my face, and just love me so much no matter what kind of person I am.
Cabin Fever: When everything goes to hell, sultry college coed Marcy tells her friend Paul that she's all but given up hope of surviving, and just wants to go out with a bang. He obliges.
In The World's End, the main character Gary King gives one of these when he reaches the final pub of his pub crawl. It details all his lost hopes and explains that he is so obsessed with finishing the crawl as it's the only time he was truly happy and he has tried to commit suicide out of disillusionment of how his life wasn't as awesome as he was hoping it to be.
Hotch also gets one to no one in particular in the form of narration at the end of "To Hell... And Back". His soliloquy about how much everything sucks is interrupted by the Reaper breaking into his apartment and trying to kill him. Yeah.
The "Life is like a box of chocolates..." monologue delivered by the Cigarette-Smoking Man in an episode of The X-Files is very cynical and nihilistic, even for him.
Dean from Supernatural delivers one of these every three weeks. The most notable one is when he admits that he wishes he'd never been brought back to life, and concludes "I should've stayed dead." Sam just mopes.
Castiel gives lots of despair speeches, but he usually combines them with threats or acts of violence, so they end up being badass or funny rather than wangsty. Except in season six, where he gives a drawn out despair speech to God. Lucifer was also especially fond of the Despair Speech, although he seemed to use it more as a tool to gain sympathy, manipulate unwitting humans, and justify his callous behaviour than as a genuine expression of despair. Luci's continual despair speeches are part of the reason that most of the other characters think of him as a bratty child.
To-morrow, and to-morrow, and to-morrow, Creeps in this petty pace from day to day, To the last syllable of recorded time; And all our yesterdays have lighted fools The way to dusty death. Out, out, brief candle! Life's but a walking shadow, a poor player, That struts and frets his hour upon the stage, And then is heard no more. It is a tale Told by an idiot, full of sound and fury, Signifying nothing.
For God's sake, let us sit upon the ground And tell sad stories of the death of Kings. How some have been deposed, some slain in war, Some haunted by the ghosts they have deposed, Some poison'd by their wives, some sleeping kill'd,...
Hope at the beginning of Chapter 11 of Final Fantasy XIII. In fact, most of the l'Cie just before their Eidolon battles.
Dragon Age II: If you side with the Mages in the endgame, Orsino delivers one after seeing so many of his fellow mages dead.
Orsino: Look at it all. Why don't they just drown us as infants? Why give us the illusion of hope?
Solid Snake gets one in the original Metal Gear Solid if you fail to save Meryl. "I'm not the hero you thought I was! I'm nothing!"
In King's Quest VI: Heir Today, Gone Tomorrow, Alexander comes up with a Despair Speech while acting like a Drama Princein front of Shamir Shamazel and the Pawn Shop Owner. Your Mileage May Vary on this, but it is the finisher in the speech, "I am... no... more..." after downing the whole "Drink Me" potion bottle that could tug at our heartstrings a bit... if we didn't know it was Faux Death.
Most of the people still alive in Bastion's Keep at the beginning of Act IV of Diablo III have these to give following Leah's death and the unleashing of Diablo as the Prime Evil upon the High Heavens, but the most poignant of all is Haedrig's.
Haedrig: I thought I could make my wife's death mean something. It doesn't matter now, does it?
Player: It does matter. You are here now, and we might still turn the tide of this battle.
Haedrig: Right. I'm sure that's a comfort for Leah now that she's gone too. I've been a fool. You don't get to make things right. This world isn't made for redemption.
Oersted gets a great one near the end of the Medieval Chapter in Live A Live.
Happens a few times in Dawn of War, mostly in Retribution.
In Dishonored, Daud gives this when you defeat him in a duel, stating that after assassinating the Empress, he felt that "something broke" inside of him and how he is filled with intense self-hatred at how rather than making something with his life he's simply chosen to use his powers as an assassin, a life he's grown tired of. At the end, he leaves his fate to Corvo's hands, seemingly not caring whether or not he lives or dies (despite his pleading for his life).
Lemongrab: No one... No one understands. I am alone! And you made me like this! *falls from window* YOU MADE ME! *runs away, ripping off his clothes* YOOOU MAAADE MEEE!!! YOU'RE... MY... GLOB! YOU'RE MY GLOB!
In the Moral Orel episode "Sacrifice", Clay gives one of these combined with a "The Reason You Suck" Speech; he rants and raves at his friends about their failures in love and life, but he is quite blatantly talking about himself too.