Green Lantern John Stewart nearly killed himself after being unable to stop the destruction of an inhabited planet (partly due to his own overconfidence). Martian Manhunter used reverse psychology to talk him out of it.
Black Hand killed himself (and his family) only to be raised as the first Black Lantern by Nekron and Scar, beginning the Blackest Night.
Nny, the protagonist of Johnny the Homicidal Maniac, regularly attempts to end his life, though he is always stopped by one thing or another. He does eventually end up killing himself... though he didn't mean to.
It was through a suicide machine designed to kill him when he answers his phone. Guess what happens for the first time ever?
Alter also puts a gun to her head in one scene. She doesn't pull the trigger because Alter wants to die in combat and therefore only a man (Yorick) has the right to kill her. Alter's self-destructive actions are motivated by Survivor Guilt, most likely over the accidental death of her sister at the hands of the Israeli military.
Wobbly-Headed Bob, a character created by Jhonen Vasquez is a Philosopher with a talent for driving the blissfully ignorant creatures of his world into suicide by telling them about how the world is supposed to be a crappy place and they're all stupid for being happy. He gets called out by this... To which he drives the critic to suicide as well. Nonetheless, it seems for the most part the suicide part is unintentional, with Bob merely trying to get others to wise up in his mind.
Sunstreaker in "All Hail Megatron". After stupidly trusting Starscream and getting himself and his fellow Autobots stranded on Cybertron in imminent danger of death, and after watching Mirage beaten up in his place by Ironhide as a suspected traitor, Sunstreaker can't take it any more and just wants to die. Which he apparently does by detonating an explosive to destroy a bridge he's on while surrounded by the mutant Insecticon Swarm.
All Hail Megatron #14 indicated that he might have survived. Ironhide #3 confirms it, though he's badly damaged and his mind really messed up. Understandable after what he's been through.
After burying Spider-man alive, impersonating him, and defeating an enemy Spider-man couldn't, Kraven killed himself because after proving himself better than Spider-man in every way, he had defeated what was supposed to be the ultimate prey and had nothing left to hunt. Yes that's incredibly flawed logic, especially in the Marvel Universe, but Kraven was insane.
Years later, Mysterio would kill himself the same way after his plan to drive Daredevil insane and do this to himself failed and Daredevil gave him a very painful "The Reason You Suck" Speech. Even worse? Mysterio was already dying and the only reason he went after Daredevil was because he didn't want to deal with the Spider-Man that was active at the time (implied to be the clone Ben Reily).
Susan Smith of Funky Winkerbean attempted suicide when she realized that the teacher she had a crush on loved someone else. She's probably contemplating it again. Also had another character commit suicide when he realized he wouldn't make valedictorian. What is it with Batuik wanting smart people to kill themselves?
Implied at the end of Ghost World. Enid feels completely isolated from everything and everyone she ever cared about; she is last seen catching a bus, a common metaphor for suicide.
In Persepolis, Marjane attempts suicide. She recovers.
In The Intimates, Dead Kid Fred attempts suicide out of depression and disgust at being a zombie. Punchy, who stumbled upon his LiveJournal and saw the warning signs, rushed to stop him, but it was later revealed that he wished he had been too late. Because then he would be a hero. That and he felt his "origin" wasn't up to snuff; his sister getting killed wasn't enough, he needed another tragedy to make him more credible as a superhero.
In Love And Rockets, Tonantzin burns herself alive outside a US embassy somewhere in the world, as a result of depression manifesting as political despair.
The newest series of X-Factor begins with Rictor standing on the ledge of a building, ready to jump, because he can't cope with the loss of of powers. He actually ends up getting pushed off by a dupe of Jamie Madrox's, but gets caught before he goes splat.
A similar situation happened to The Blob after the loss of his powers. He tried to slash his wrists, but was unable to find a vein amongst all the stretched out skin.
Minuteman Jack "The Monster" Daw from 100 Bullets. He's addicted to self destructive tendencies: alcohol, heroin, and violence. He makes remarks about wanting to die yet he teases death with the needle or the gun as Graves puts it. The gun being a metaphor for violence. Jack got clean but he hasn't really changed since he keeps seeking the kick (violence) that diverts him from facing his pain. There is a moment where he avoids violence and seems to be getting better. Jack "relapses" (violence); kills a guard by squeezing his head with his foot followed by challenging candidate-minuteman Crete. Jack dies fighting crete when alligators devour them both. It shows us that he stayed married to his self destructive nature until the end.
Minuteman Milo "The Bastard" Garret. He earned that nickname by being the most ruthless of the minutemen. In sleeper mode he worked as a private detective and when he was reactivated he didn't like who he was. Disguised in bandages he commits suicide by Lono, taunting and challenging him, pulling his famous knockout punches so Lono would kill him.
In X-Men Noir, Warren Worthington jumped off the roof of Professor Xavier's reform school after learning the truth about Jean and just how twisted she really is. The X-Men are convinced the police did him in so they'd have an excuse to arrest Xavier; the police are convinced Xavier's tutelage drove him over the edge.
Sharon Ventura, one of The Thing's love interests, attempts this twice - once after she becomes the She-Thing and again after the Thing ends up rebuffing her advances when he learned that she nearly allied herself with Dr. Doom.
In the Teen Titans spinoff Vigilante, after the deaths of several friends attempting to take up his mantle during a period of retirement and being unmasked on live TV lead to him becoming more angry, violent, paranoid and obsessed with dispensing justice not caring if he murders even innocent cops who get in his way, Adrian Chase (the title character) reaches this point by the end of the series. He succeeds as the final issue ends with Adrian shooting himself in his apartment.
Carl Barks' "Dangerous Disguise" is probably the only Disney comic book story to ever show a character taking his own life - when a foreign spy fails his mission and realizes that his totalitarian leader will now send him to salt mines, he chooses to finish himself by jumping out of window.
Several characters in "The Walking Dead", including Hershel Greene, Maggie Greene, Chris, Julie, Douglas, and Carol. Carol is the only one to go through with it unassisted.
In W.I.T.C.H., Prince Phobos ends up throwing himself off the edge of the Citadel of Kandrakar after his last plan blows up in his face. The fact that there's nothing beyond the Citadel means he'll be falling for a long time... so he's probably dead.
X-23 habitually cuts herself on her wrists and forearms with her claws, and it's hinted that sometimes she is inflicting fatal wounds on herself when she does so. The only reason she hasn't died from them is because of her Healing Factor. Target X more blatantly indicates Laura may be prone to bouts of suicidal depression: At some point after she is forced to cut off contact with her remaining family, she seeks out Wolverine, her genetic father, fully intending to kill them both over their part in the Weapon X project because by this point she has been so broken by everything that's happened to her she feels the only way to end it is death. Logan manages to talk her down, but her continued willingness to sacrifice herself (such as when she is infected with the Legacy Virus) suggests she may still harbor suicidal tendencies.
After his episode of clinical depression at the end of The Childrens Crusade, it was fairly obvious throughout the start of Young Avengers Volume 2 that Wiccan wasn't 100% okay. In issue 5, however, having had Loki trick him into giving him his magic, Billy takes Loki's words to heart and decides to kill himself in order to reverse everything that he'd done thus far, his internal monologue making it clear that he has no self-esteem and an overwhelming sense of guilt. Loki has a change of heart and saves him just in time, and Billy decides to don a new hero costume and deal with the problem through other means.
In Justice League Elite, after watching one of his few friends, Manitou Raven, take the full force of a bomb that was headed his way, an utterly depressed Major Disaster tried to kill himself by slashing his wrists in a bathtub. Luckily, he bungles the attempt and decides that getting sober would be a better way of paying tribute to his friend.
During the initial Runaways series, after a vampire managed to seduce her and force her to fight her crush, Karolina Dean attempted suicide by offering herself up to the vampire in exchange for his sparing her friends. It failed spectacularly because her blood was loaded with solar energy, and thus the vampire was burned alive, but she didn't know that that was going to happen...
Another proof that Disney comics can do this, too: In Paperinik New Adventures, Grrodon is the last Evronian on Earth... and he has been for the last three hundred years. When he fails to transform the hero into a Slave Mook in the future in order to change the past (it's Time Travel. It's complicated) he decides he can't keep going on, so he steals a flying car and travel to space, knowing that the Explosive Decompression will destroy the car and kill him.
Later we have Xadhoom, who kills herself in order to become a sun to allow her people to survive.
Mr. Sensitive of the X-Statix was Exactly What It Says on the Tin - his powers caused him constant agony that was only kept in check by his costume, which had to be redesigned every few months because his sensitivity kept growing. Plus, he was in an extremely high-stress job as the leader of one of the most image-conscious superhero teams in the Marvel Universe. Naturally, he contemplated suicide several times, and even kept a revolver, just in case.