Lexx from Alien Dice tried to kill himself several times but survived because of tiny machines injected into his circulation. He doesn't dare to try the most radical methods for fear of the pain if the machines would still repair his body. Only after several attempts to die failed did he gather the determination to try and win.
Justin from El Goonish Shive seriously contemplated suicide a few years ago (in comic time). Big Bad Damien actually committed suicide upon realizing he was not a god.
Luna Travoria from Dominic Deegan tried to kill herself twice in the same story arc. The first attempt ended with Luna and Dominic uncovering a plot from her crazy mother, in which she would make a great deal of money by Luna driven to suicide. The old lady would wind up dead anyway, after picking a fight with a Royal Knight and getting her head chopped off. The second time, Luna jumped off a bridge after said crazy mother left her nothing in her will, and her half-dead body was discovered by Dominic and his cat Spark while on a fishing trip.
Nimmel Feenix later tries the jumping-off-a-bridge strategy for himself after a rather horrific attack on his class by a Psycho for Hire. He's stopped by none other than Luna herself.
Questionable Content has Faye, a girl with obvious issues. It is later revealed that the reason she is so severely messed up is that her father, who she saw as her best friend and in whom she confided everything, committed suicide for reasons unknown by blowing his brains out with a gun, in front of her. He went outside in the early morning without realizing that she had gotten out of bed and followed him outside. Then he blew his brains out before she could call out to him. Yikes.
It's also implied that the car accident she was in just before the story started was an attempt. Faye says that she isn't sure herself, because she doesn't remember the moments leading up to the crash.
Artie Crowley of Concession was seen holding a gun to his head after realising he had had sex with a prepubescent boy while in a trance and unable to realise what he was doing, but he didn't go through with it.
ApenShephard of The Silver Eye lost his will to live and tried to take his own life when he was only ten years old after being exiled from his country, and a number of other tragic events.
After untold amounts of emotional suffering due to his robotic form, the title character of Warbot In Accounting finally decides to kill himself by jumping off his office building while leaving behind a note scribbled "No Love". It doesn't work due to him being Nigh Invulnerable and he ends up taking crap from his co-workers for skipping work.
Or so the BLU Medic claims; subsequent events suggest another possibility.
Otter Soldiers: Subverted. One of the characters at one point contemplates upon committing suicide. She ends up thinking too much about how it could end up going horribly wrong (survived but crippled etc.), and is ultimately unable to go through with it.
Neilli of Juathuur is driven to suicide by Meidar, who decides that a girl that wants to be a healer would make a perfect torturer. Possibly a Secret Test of Character directed at Neilli's boyfriend Thomil.
Daichi Takana's mother is apparently driven to suicide by a Mafia boss, who just so happens to be his friend's father in The Mitadake Saga
Conor from FreakAngels shoots himself in the head, not out of a true desire to die, but out of a wish to upgrade his mental abilities. His action fits this trope because he really didn't know whether he'd survive, and his conversation with Arkady and relationship with the other Freak Angels suggest that he didn't much care.
A Running Gag across MS Paint Adventures as a whole is people being driven to suicide near a stump in an empty field, later named the Land of Stumps and Dismay in Homestuck. First it was the protagonist of Jailbreak, then later an audience member of Problem Sleuth and Homestuck gives it some serious consideration in response to events in the comics themselves.
However, a few of the characters have attempted suicide: the Ace Dick who played the Game of Life with Death in Problem Sleuth, and the Handmaid in Homestuck. Both cases proved temporary, although the Handmaid eventually succeeded in Suicide by Cop.
№ 4 from Hell(p) died by suicide as a result of having a crappy life, but since the story takes place in Hell and everyone is Dead to Begin With, we only find out about it as a part of his Backstory.
Depicted tragically in story #5 of The Bully's Bully when one of BB's friends is bullied until he can see only death as an escape.
Dani from Between the Lines was sent to a psychiatric hospital after cutting her wrists and nearly dying. Being a bullied, closeted trans girl with Abusive Parents drove her off the edge.
Kotomi from Cross Heart is traumatized and haunted by the fact she was raped several years prior. She has a scar on her wrist from trying to kill herself in the past. Surprisingly her girlfriend never noticed it until three years after they began dating
Kasem from Tales of the Galli attempts suicide after being forcibly castrated, but is saved by the goddess Cybele.
A regular occurrence in Eight Cicadas. Shark is one of the biggest examples so far in the story, as he fails twice with overdosing before ultimately hanging himself in his aunt's basement.
In Evangelion 303 Asuka tried to shoot themselves in the head after several months of lashing out at everybody in Survivor's Guilt fueled rage (Shinji stopped her). It's also implied during an argument with her subconscious that she had attempted to overdose on pills in the past.