This faction forms an unholy trifectera with the Fated and the Xaositects as the Factions you're least likely to see DMs actually allow anybody to play. This is because these factions are all magnets for That Guy.
The problem with the Doomguard is that they're nihilists. Not in the deep "there is no true meaning to anything, so you need to find your own reasons for living" sense of the Bleak Cabal, but in the shallow "everything is going to fall apart! Destruction is going to win over everything, and that's awesome, because wrecking shit is cool!" sense.
That's not to say there aren't some sophisticates amongst them. Of the three distinct subgroups amongst the Sinkers, there's both a group that actually thinks that the Doomguard should be taking steps to slow the rate of decay, as too much destruction too quickly stops being entropy and starts being bedlam, and a group who prefers to take a more long-term approach. As these guys see it, not only is the natural entropy of, say, a river eroding its banks beautiful, but construction actively feeds entropy; to build a bridge, you're mining out the guts of a mountain, chopping down a forest, and working folks ever closer to death, all for the sake of making something that'll fall apart far quicker than the mountain or the forest would.
But, the majority of Doomguards want to help hurry entropy along, especially since this is the active goal of the Factol, a 20th level Chaotic Neutral female planar human Ranger named Pentar, formerly of the gate-town of Xaos on the Outlands. In fact, although she's predominantly focused on training to try and throw a monkeywrench into The Great Modron March, she takes a secret delight in the growing seeds of rebellion in her own faction, hoping to push it towards some entropy of its own - maybe it'll even fall apart!
Needless to say, this doesn't make the Faction a lot of friends in-universe. In fact, the sheer amount of chaos they caused during the Great Upheaval almost got them stripped of Factionhood altogether - if it wasn't for Factol Skall standing in opposition, that's exactly what would have happened. To this day, only two Factions can really be said to like the Sinkers; the Bleak Cabal (who find common ground in decay) and the Dustmen (who like the idea that "False Life" itself is destined to die). All of the more lawful Factions despise them, although none more so than the Harmonium.
Now, you'd think that this would cement the Doomguard as a grounding point for Chaos and Evil on the old alignment grid, and it's definitely true it's attractive to a certain flavor of both Stupid Evil and Chaotic Stupid, but the truth of it is that all alignments are welcome and present in the Sinkers. If you're trying to wrap your head around how that works, essentially, it's based on how the cutter actually feels about entropy and how they go about supporting it; chaotic individuals typically gravitate towards accelerating entropy and lawful ones towards slowing it, whilst good Sinkers prefer to support entropy through inaction (for example, stopping others for shoring up a decrepit building) and evil ones through action.
The only real restriction is that a cleric with access to the spheres of Creation and Healing can't become a Sinker. The magics they wield are a slap in the face of the concept of entropy.
Because of their beliefs, the Doomguard have taken over the Armory - whereas the general metalworking of Sigil is done by the Believers of the Source at the Great Foundry, the Armory is the center of the weapons trade in Sigil. You can buy every sort of weapon here, enchanted or mundane, from the most humble to the most exotic. Want a dagger of blood obsidian? Want a demon-possessed cannon that fires screaming skulls? Want a tunnel-sweeping ball of motorized bladed death? You can buy it at the Armory, if you can afford the price.
Actually joining the Doomguard isn't as easy as you might think. They love entropy, but they don't want to be just a haven for random shit-stirrers. As a result, a would-be Sinker needs to approach one of the higher-ups at the Armory who'll assign them three tasks. The first: smash one of their own weapons to pieces on the outer walls of the Armory. Second, take a sack of at least 500 gold pieces and scatter it through one or more public places in the Hive Ward - with definite points for sparking a riot. Finally, and this is the tricky one, prevent the dabus from trimming back the razorvine on any single overgrown building for a full day. As killing a dabus will almost certainly get you mazed or flayed by the Lady of Pain, this test ensures that those who get to wear the Doomguard symbol have dedication and brains to go with their love of ruination.
The Doomguard are, as you might expect, deeply tied up with the Negative Energy Plane. Not only do they maintain mighty citadels on each of the four negative quasielemental planes (Ash, Dust, Salt, Vacuum), but the greatest champions of the Doomguard are the so-called Doomlords, of which the four most powerful are incredibly powerful unique undead. Their biggest secret is that the four towers of the Armory are actually gigantic portals; each tower exists simultaneously in Sigil and one another plane, where they form the aforementioned citadels.
Needless to say, Doomguard are a fairly combat-focused set of bashers. All Doomguard get proficiency with the sword, even those classes normally denied it, as well as the power to deliver a deadly Entropic Blow once per week. Additionally by touching destroyed material, if the destruction happened at some point within their sensory range (which increases with level), a Sinker can get a psychic impression of what destroyed it. Sinker Priests can do the same thing with organic remains as well.
The drawback, of course, is simple: Doomguard are not allowed to heal themselves with magic. This is so strongly ingrained into them that any injured Sinker automatically attempts to make a save vs. spell whenever a healing spell is cast on them, nullifying it if they pass.
Needless to say, the Faction War doesn't do this faction much good. Duke Rowan Darkwood plays them for suckers, manipulating them into starting a war with the Harmonium - not helped with the Lady of Pain tries to pre-empt the mess by mazing both Pentar and Erin Montgomery, Factol of the Sensates. Come day 10, Pentar's most important underling Ely Cromlich is giving out Doomguard-forged weapons for free to anyone who promises to use them to fight the Harmonium, which comes to a boil on day 25, when the Harmonium, Sons of Mercy and Sensates attack the Armory, which is defended by the Doomguard, Revolutionary League, Indeps, and Sodkillers. The result is the destruction of the Armory and massive casualties for the Doomguard.
Needless to say, when it's all over and done with, the faction is pretty much wiped out. Almost every Sinker is dead, the survivors are despised by the denizens of Sigil for starting the war (even if they really were duped by Duke Darkwood), and so they abandon Sigil and flee for their citadels on the Inner Planes. Of course, as the very philosophy of their faction demands they allow it to dissolve into nothingness, the Doomguard are as good as finished.