Deadshot, an assassin in The DCU. His death wish stems from an incident in his childhood; his wealthy parents hated each other, and mom tried to hire her sons to kill their father. The younger brother, Floyd, tried to prevent his older brother from doing so, but his brother kicked him out of the house. Floyd got his hands on a gun and stood on a tree branch, hoping to simply wound his brother, but the branch snapped and he missed. So he killed the brother he loved to save a father he hated. Eventually, Floyd became an assassin for hire, assuming the Deadshot identity. After multiple stints on the aptly named Suicide Squad and a few fights against other badass assassins, he's still alive.
Mr. Immortal of the Great Lakes Avengers is named for his one power, which is to come back to life after dying. He discovered this after his girlfriend Terri killed herself and he tried to follow suit. After dozens of suicide attempts, he finally realizes what's going on (little slow on the uptake) and decides to fight crime. However, he is still prone to depression (especially after his new Love Interest Dinah Soar was killed) and occasionally goes on multiple suicide sprees. Since he now knows about his powers, it's unclear if he is actually looking for a loophole, or if it's just his way of blowing off steam.
Yorick of Y: The Last Man is like this for the first few volumes, thanks to Survivor Guilt after The End of the World as We Know It. It turns out that this is also Alter Tse'elon's motivation for everything she does in the series — she's trying to die in honorable combat with a man.
Morpheus. If he indeed was (his methodology makes it somewhat uncertain how much was planned and how much was not), he certainly went about doing it in anextremelyroundabout way. Further backed by Word of God. Gaiman once summarized Sandman in the sentence "The King of Dreams learns that all things must change or die and makes his choice."
Another story concerned a female counterpart of Metamorpho, who was Driven to Suicide by her hideous form but whose body automatically protected itself from any attack. Naturally, the story features Death.
The anthology Endless Nights had a soldier whose life was empty, until he encountered Death and actually helped her on a job by breaking down a magic gate so she could enter a castle and claim the inhabitants. Now, he enters battle with renewed vigor in the hopes he will meet her again.
Daredevil (not Matt Murdock) of Earth X is unkillable due to his regenerative powers. It's not clear how he became suicidal, but he becomes the center of a circus act in which the audience is invited to kill him if they can. Later he tries to get several Big Bads to kill him, culminating in his multiplying into many versions of himself when he is ripped apart. Late in the series he apparently finally gets his wish, but only after all of humanity has joined him in painful immortality (due to the death of Death).
Batgirl III's guilt over reducing a living, thinking creature to a large mass of inert meat with her bare hand at eight years old drove her to seek her own demise protecting others. It can be summed up when Lady Shiva demanded a duel to the death in a years time as payment for helping her regain the body-reading skills she lost when a telepath enabled her to speak:
Cassandra:[thinking] I'll never take another life, not even hers. So I'll pretend to go all out, and then I'll die. I don't have to do this. I can still study Batman's method. I can be good enough for the costume. I can be...mediocre. For a lifetime. Or perfect...for a year. Shiva: Well?
While it's not explicitly stated Batman is constantly reflecting on the life-and-death situations he finds himself in and musing that "this would be a good death," the clear implication is that he's actively looking to go out in a suitable blaze of glory. Ultimately subverted; although Batman appears to do just that while fighting Superman, it's revealed that he faked his own death, having discovered a new purpose to live over the course of the story.
When Harvey Dent publically reveals that his face has been surgically restored, he appears to have turned over a new leaf. However, Two-Face-related crimes begin happening not long after, and Batman speculates to Gordon that Harvey may be looking to die. Sure enough, in the climactic confrontation in the first volume, Harvey nearly falls to his death trying to shoot Batman, and after a last-minute save, it's revealed that he's gone even more insane, seeing his face as completely destroyed, and has no will to live anymore.
In Batman Vampire, Batman, after spending Bloodstorm as a Vampire Refugee, has succumbed to his vampire instincts and embarks on a killing spree, slaughtering most of his Rogues Gallery for their blood. During one of his private monologues, he ponders whether or not he's killing to satisfy his bloodlust or to provoke Gordon and Alfred into taking action against and killing him.
Clyde Rawlins in Robin Vol 1 has abandoned his previous life and gone rogue and is explicitly looking to go down fighting and take King Snake with him for revenge. Clyde does find his death but he does little damage to Snake in return.
Despite Tim's introduction into the Bat mythos as helping Bruce out of this by the time Tim has switched ids to Red Robin he is being accused of falling to self-destructive tendencies himself, and his responses are telling non-answers. Cassie and Dick are given two different flavors of misdirection, but it's the way he brushes off Steph rather than answer her that seals the deal, since she's on the short list of people he wouldn't feed that type of misdirection to.
This, not vengeance against criminals like the ones who killed his family, has been The Punisher's true goal his entire career. That's the reason he's so reckless and cares little about the consequences of his actions. He hopes that someday he'll be killed in action so that he can finally be at peace, but if he can take as much scum off the street as he can before that happens, all the better.
Hank Henshaw, aka the Cyborg Superman, became one of these, tired of the tragedy in his life and his near invulnerability. It got to the point where he joined the Sinestro Corps just because they agreed to kill him once their work was done. He eventually did die, much to his delight. Unfortunately for him, his minions resurrected him at the first opportunity. The first thing he did was shed a Single Tear over being alive.
The Mighty Thor: Eilif the Lost was the last survivor of a Viking Lost Colony in Antarctica. Old and growing infirm, he tried to goad Thor into killing him. "I would have fought a god, my lord. What Viking could have asked for a more glorious death?"
Dashiell "Dash" Bad Horse from Scalped has an unconscious death wish. He constantly throws himself in dangerous gunfights with psychopaths and always alone. It's hinted that he suffers from PTSD (child abuse, fighting in Kosovo and witnessing a massacre), suicidal behaviour (flashback to a young Dashiell cry and put a gun to his temple, second time he does the same thing when he could've prevented the murder of a young boy) and from deep seated anger (his fists are bruised most of the time and we see him slamming his fists against his own truck). When Dash is confronted by his father, the conversation between them confirms all of this and more importantly his death wish.
Kaine, the imperfect clone of Spider-Man. He's not the typical death seeker as he sometimes changes his mind or even runs from a fatal fight. In the Grim Hunt back stories it's revealed he's too much of a coward to commit suicide, yet when he has a pre-cog vision about his own death at the hands of Kraven the Hunter he still challenges him.He finally got what he wanted by duping the Kravinoff hunters into thinking he was Spider-Man. They sacrificed him to revive Kraven the hunter. Afterwards, Kaine is resurrected with tarantula features, and is ultimately reborn possessing Spider-Man's enhanced powers from Avengers Disassembled.
Dara Brighton in The Sword insists that she is already dead after the murder of her family. She just wants to hunt down and kill the demigods responsible for said murders before she actually dies. She does and she does.
Lady Shiva of The DCU is a Death Seeker like Cassandra Cain her own daughter mentioned earlier. Shiva has always regretted that her sister was killed for the sake of her own potential as a martial artist and confessed to Cassandra that she misses her every day. As a result, Shiva is a mix of a Blood Knight and a Death Seeker. She continues to challenge and train gifted martial artists out of a need to validate her sister's death by proving that she is the strongest, but at the same time, she secretly hopes to die at the hands of someone better due to her guilt.
In one Star Wars story arc, the crooked ex-Senate Guard Venco Autem learns that he has a terminal illness, and so takes on suicidal jobs like assassinating corrupt Senators because he has nothing to live for. At the end of the comic, he places himself in a situation that he has little hope of escaping from in order to kill the Senator, and is indeed shot dead by his brother.
Deadpool wants to die. Death is in love with him and vice versa, so jealous Thanos cursed him with eternal life. In the Age of Stryfe, an alternate future timeline Deadpool is still alive because of the curse and not his healing factor. The voices in Deadpool confirm his deathwish among other things: the only reason he desperately wants to be accepted is so his friends will care and put him out of his misery. If that doesn't work out he antagonizes them so they can exact their revenge on him. The later "Dead" storyline finally gives this to him via Spider-Man villain Tombstone. However, it doesn't stick - he comes back, but he loses his healing factor, but his body is totally healed and he's regained his true face. More importantly, he doesn't want to die anymore.
Sadly, this has come back for poor Wade. After the events of Secret Empire and the villain Stryfe forcing him to try to kill his friends, he's put a $20 million bounty on his own head with the hopes that someone will finally murder him as he feels he's not worth it anymore with all of his credibility shattered.
Kraven's Last Hunt: Kraven the Hunter's main reason for hunting was to escape the harshness of life: "in the midst of pursuit...or a kill...I do not exist..all the petty pain of living falls away..all unrealized desires...until only the hunt is left". In the end, he killed himself with a rifle in his mouth but he was resurrected years later against his wishes by the remaining kravinoff members. The ceremony of resurrection was corrupted so Kraven cannot die. Only the antithesis of a hunter (the spider) can kill him, so he hunts down spiders. Looking for a way to die with honor he went after Spider-Man but he refused to kill him.
The events of Target: X (and possibly NYX, the timeline isn't clear) leave X-23 in this state, and she tracks down Wolverine intending to kill both him and herself in order to put a permanent end to the Weapon X project. Logan talks her down, but even after Laura has shown a very poor sense of self-worth and a suicidal disregard for her safety, constantly putting her life at risk to protect others (such as taking a full-force blast from Nimrod that otherwise would have struck Hellion) or otherwise expressing a willingness to sacrifice herself (IE, taking the Legacy Virus into herself and then intending to commit suicide without a second thought. Even though Elixir, who could completely eradicate the virus from her body when her Healing Factor failed to stop it, was right there, and ultimately did end up healing her before she could kill herself). Her solo series has helped her learn to value her life, but she at times still struggles, and it's not helped that every time she seems to be turning the page on the pain in her past, something happens to threaten everything she's rebuilt.
In Action Comics, Doomsday the monster that nearly killed Superman becomes this millions of years in the future. Filled with self-loathing, he has gone back in time to remove every last trace of himself from the past, present, and future. Since people on Earth know how to clone him, he figures he needs to get rid of Earth as well.
In The Goon comics the immortally weary Buzzard agrees to go kill a monster terrorizing the countryside, hoping it will be able to kill him. When he arrives he learns that the ancient creature was doing this with the hope that they would fight back and kill it. They agree to fight it out and let fate decide which one of them...wins.
God in the Fallen Angel series. The Alpha is ready for the Omega, complete oblivion. He's being denied the ultimate rest because people keep praying to him. Enraged he punishes mankind with all kind of disasters so they will finally realise that their god does not care and so he can finally sink into the sweet surrender of nothingness. He refrains from destroying the human race entirely because he doesn't want to admit that his creation was an utter failure. If the worship eventually doesn't stop, he is not above omnicide so he can finally get some peace and quiet.
Red Hood and the Outlaws: Arsenal fights Killer Croc for the sole purpose of dying looking like a hero, after hitting rock bottom.
The Talon implies that Red Hood may be one to some degree (especially considering how the fight was handled as a Not So Different situation).
The Talon: I want to end this "life" on my terms. Sure you can understand that.
In Runaways, after losing his girlfriend, Chase Stein tried to kill himself by offering his life to the Gibborim in exchange for Gert's resurrection. They weren't interested, because his soul wasn't pure, but they used him to draw in the other Runaways so that they could try and take Nico. Thankfully, their plan failed, and Chase decided not to kill himself after all.
In The Spirit story "Wild Rice", Rice Wilder ran from her wedding into the arms of a bank robber after years of rebelling against the Gilded Cage of her family's wealth. In the end, when the Spirit offered to take her home, she broke away and made a run for it, only to be shot by the bank robber who thought she had betrayed him to the cops. She died with a smile on her face, her last words being, "I'm finally free..."
Jack Russell a.k.a. Werewolf by Night comes to suffer from his curse so badly that he reaches the point he no longer cares whether he lives or dies. He signs up for what will very likely be a suicide mission, which results in a fight with best friend Morbius whom he tells to do him a favor for once and just kill him.
The Joker is sometimes implied to be one, his monstrous acts being attempts to court death. Over time it's evolved a bit; he still wants to die, but he now wants Batman specifically to be the one who does it. Batman: Arkham Origins explores this in more detail. When he first arrives in Gotham, he's obviously expecting/hoping to get killed by the numerous people he's provoked. His obsession with Batman comes about when Batman refuses to kill him and even saves him despite the horrible crimes he committed; Joker is so confused that he becomes enamored with Batman, viewing him as his perfect opposite.
In the Joker graphic novel, Joker's henchman Jonny Frost seems to be one as in one scene during the story's midpoint, he is standing on the roof of his apartment building, looking down with a sad expression on his face. He finds it during the story's climax, jumping off a bridge after being shot in the face by the Joker.
In The Transformers: More Than Meets the Eye, Chromedome certainly seems less than eager to continue living. During the War, he was actively suicidal, to the point where he met his current conjunx endura, Rewind, at an assisted suicide clinic. In the plot, he has continued to engage in mnemosurgery despite its tendency to slowly kill its practitioners, and he got so reckless when performing it on a near-unstoppable captive Decepticon Super Soldier that he stopped telling anyone he was going in. When he finds out that Rewind's former conjunx endura has been Mode Locked as the Pet of the DJD, he attempts to bring back Dominus at the cost of his own life; Rewind has to sever his arm to get him to stop.
Whirl was showing these warning signs as early as the first issue, where he may or may not have been about to commit fiery suicide atop a pile of mutilated Sweep corpses. Throughout the series, he spends his time picking fights with people and generally being a Jerkass, and never misses an opportunity for battle, apparently in the hope that someone will kill him; when he's held at gunpoint by a Shell-Shocked Veteran who's having a disastrous episode, he demands that Fortress Maximus pull the trigger. Amusingly, when the party does seem to end up in the afterlife, Whirl decides that it actually kind of sucks and starts needling Rodimus about getting out again.
Whirl: If I'd known death was going to be like this, I wouldn't have courted it so much. Angry denials, stunned silences. Gasps of disbelief. It's like that time Magnus tried to tell a joke.
The entire plot of Druuna: Morbus Gravis is initiated because Captain Lewis is sick and tired of being unable to die and existing as nothing more than a disembodied head plugged into the ship's systems. He tricks Druuna into believing that deactivating the Master Computer responsible for running the massive city-ship, which has gone rogue, will solve the Viral Transformation plague that is wrecking it. It instead triggers the self-destruct, but Lewis ultimately can't bring himself to let Druuna die along with the rest of the human race simply because he loves her so much.
In the graphic novel version of Myth Adventures, this is Isstvan's goal after being cursed with immortality. Everything he does as a villain is in pursuit of this goal. The novel never gives him a real motive besides a hunger for more power.
Eternity Girl is a comic-book miniseries about a washed-up old superheroine called Chrysalis who's been trying to kill herself for years, but has been thwarted because her own superpowers make her nearly impervious to any injury.
The Ultimates: After losing his family, he lost the will to live, and kept fighting just as an excuse to eventually get killed. Such as picking a fight with Wolverine, for really no reason...