"Consider it: every person you have ever met, every person will suffer the loss of his friends and family. All are going to lose everything they love in this world. Why would one want to be anything but kind to them in the meantime?" -Sam Harris
"Life's a bitch and then you die."
One of the more out-there Factions, and probably something that would have been written better if Planescape had come out a couple of years later, the Bleak Cabal's philosophy is essentially one of nihilism; the Bleakers cement themselves around the shared belief that there is no "great plan" to the multiverse, that there is no "purpose" to anything. This is similar to yet different from the "the universe is random" philosophy of the Xaositects or the "everything exists to fall apart" philosophy of the Doomguard; it cements itself around the idea that the only meaning the universe has is what a person chooses to give to it.
This is seen as a very bleak and grim philosophy, hence the name, and is reflected mechanically by the idea that Bleakers are at severe risk of falling into melancholy-induced catatonia called the Grim Retreat because they have no "higher meaning" to hold onto. As a result, active proselytizing isn't something they place a lot of emphasis on; most people are drawn to the Bleak Cabal of their own, rather than coaxed into joining. By this, we mean that entry into the Bleak Cabal isn't as simple as just walking up to one of their soup-kitchens or the Gatehouse and saying "I want to join". Okay, technically, it is that simple, but the thing is that no Bleaker will pay the prospective member any attention. At most, they'll occasionally try to dissuade them from joining. Only a soul who can tolerate this apathy-hazing, which lasts an average of 6-12 months, will be accepted and recognized as a namer, which requires dropping all parts of their name but their given name, to symbolize their rejection of meaning.
At the same time, though, the Bleakers are one of the most sincerely altruistic factions in all of Sigil; if there is no reason to be good and kind, then that means it's all the more important when you choose to commit kindness, because you're making the choice to do so. The Bleakers operate the sanitariums, hospitals and soup kitchens of Sigil, struggling constantly against the indifferent cruelty of the city around them to make the lives of the poor a little less miserable. The current Factol of the Bleakers, a half-orc named Lhar, is planning on greatly expanding the facilities of the Faction to prepare for what he believes to be a great influx of Bleakers who undergo the Grim Retreat, as well as build a political network for the normally apolitical Faction. After this Lhar plans to succumb to the Grim Retreat. However, he is fully intending to return from it, which would make him the very first person to do so. To this purpose he has enlisted four members of the Faction to see him through this approaching madness.
After the Faction War, the Bleakers simply shrug their shoulders, say "okay, we're not a faction", and just keep on working the same as they did before.
Because they're already so inured to being on the edge of madness from all their existential bleakness, Bleakers are immune to madness-inducing spells and to certain psionic abilities that have a similar nature, such as ego whip, psychic crush and psychic surgery. Furthermore, after reaching 7th level, a Bleaker can attempt to alleviate artificially induced (as in, caused by magical items, spells or monster abilities) madness.
This is a long process and not without its risks. First, the Bleaker needs to spend an hour meditating in order to properly clear their mind - they only have an 85% chance of success at this, and if they fail the check twice for the same target, then they can't ever use this Absorb Madness power on them. Then, they spend d12+4 hours ritually tending to the restrained barmy, and if the ritual is interrupted, it's got to be started over. If completed, though, the victim is permanently cured but the Bleaker becomes incapacitated for the next two days as they expunge the absorbed insanity.
In addition to that, Bleaker fighters and rogues of 10th level or higher have a special requirement, but which gives them a special benefit in return. They have to donate 50% of all treasure they earn on their adventures to charity, but they gain +1 Charisma point per 2 levels as a result, ignoring normal racial maximums and allowing them to reach Charisma 25. At which point they're automatically granted the best possible outcome in any meeting with those of opposing viewpoints.
Spellcasters, meanwhile, are taught two unique spells developed by the Bleak Cabal; the 4th level Enchantment spell Despair, which crushes the viewpoint of others by exposing them to the Bleaker's worldview (for this reason, it's not as effective on primes or members of the Free League), and the 6th level Conjuration spell Howl of Pandemonium, which summons screaming winds from the wizard's mouth that deafens, deflects missiles, negates sound-based attacks, physically repels those trying to approach the caster, and can potentially drive folks temporarily loopy.
Faction drawbacks? A daily D20 roll to avoid succumbing to despair, the potential to end up in the Gatehouse as a result of that, and a sizable loss of lifespan depending on race - everything from 10 years for humans and tieflings to 400 years for gray elves.