IonFrigate's Abyssal Knight guide
Note: This guide was originally written as a reddit post, found here. It has been reproduced here so that it can be more easily wikified and linked to, and so it will appear in the relevant categories. It was written with 0.10 in mind, and from experiences with 0.8 and 0.9. Some parts are out of date, and will become even more so when 0.12 becomes the stable version. However, most of the general advice on handling Lugonu characters in the main dungeon is reasonably current as of 0.11.
The Starting Abyss
This is a really annoying bit of window dressing. It looks like crap, and it's not like the normal Abyss. The items you find are D:1 floor trash, but the monsters are the same D:20-27 types you see in the normal Abyss, and there's no chance of finding the Abyssal rune. Obviously, under these circumstances, you really just want to get out. You can use the ability to do that immediately, but if you wander a bit, you can usually find a gate out, and save yourself the piety. Note that if the Abyss shifts, this is no longer true: just gate yourself out.
The Early Game: Kill and Banish Dudes
Very early on, your +2 weapon should be enough to deal with D:1 nonsense. Use Bend Space to save yourself if necessary, but don't spam it - it damages you. Sacrifice everything you don't need to eat, in order to get up to *** piety. Then, Lugonu's power really starts to shine through: you can banish monsters to the Abyss. This instantly makes them gone from the level forever, though you may encounter them again if you visit the Abyss. Be aware that the invocation can fail, and that monsters can resist it. Also note that it costs piety every time it's successfully cast, even if the monster resists. This means banishing some difficult opponents can rapidly drain your piety, but in many cases, this is worth it.
But do be careful, since banishing monsters comes with several downsides. The first, as mentioned above, is that it drains your piety. The second is that you lose experience. In 0.10, you get XP proportional to the percentage of the monster's max HP you had already inflicted as damage on it. So, if an ogre has 30 HP and you banish it after inflicting 10 points of damage, you get 33% the XP you would normally get. In trunk this has been upped to 50 + damage/maxHP percent, so for that same kill you'd get 67% XP. The last downside is you lose for the time being whatever items that monster was carrying. Usually monsters only carry trash, but some monsters, particularly uniques, carry good stuff.
On Skill Training
This depends so heavily on what you want to do. It is possible to play AK as straight fighters, meaning that you'll pretty much train the standard skills: a weapon skill, Fighting, Dodging/Armour, Shields, etc. On the other hand, if you're playing one of the more casty races, like DE or HE, you'll probably eventually want to cast. When and how you start depends heavily on what books you find, and I can tell you this isn't a strategy for beginners, though it's far more powerful in the end (Lugonu's abilities go very nicely with a conjurer build).
But one thing holds true for all Lugonu characters, though: they all need to train Invocations. You want to be able to use Banish reliably and at a good power, and you want the Enter the Abyss ability to actually be a reliable lifesaving technique. Train it until Banish and Enter are at levels you're comfortable with. For this reason, HO make good AK, given their strong Invo apt (+2) that's unaccompanied by other bullshit (unlike, for example, DD or undead).
On Distortion Weapons
These are neck-in-neck with electrocution weapons for the best brand in the game. They do a set amount of extra damage on average, unlike most other brands whose damage is based on the weapon's base damage; this means it's best to have them on fast weapons (those with a base delay of 13 [130%] or less). By far the best is a quick blade, with a minimum delay of 3. Unfortunately, they are very rare, and you'll probably need to burn several scrolls of acquirement to get one.
Distortion weapons come with two major downsides, though. The first is that you can't unwield them without risking banishment, glow, confusion, HP damage, and other things. Lugonu helps only against the banishment; the other effects will be just as nasty. In effect, this means you can't change to a better weapon readily, so it throws launchers, staves, and rods out the window as options. It also means that if you have a blunt distortion weapon, you can't butcher. This can be mostly worked around with Animate Skeleton, which is easy to cast but not always found.
The second downside is simple: distortion weapons are annoying. They teleport monsters you were about to kill to the other side of the level where they heal, they blink centaurs around you giving them a few more shots with their bows, and have a tendency to banish monsters, denying you items and some experience. This is bearable, but it definitely makes the weapons less attractive.
Also, note that blink frogs are immune to extra damage from distortion weapons. Not that this really matters, since they're pretty easy to banish (see above) or take down with wands.
As for when to get it? That's entirely your call. It somewhat depends on what you want to do with it. Do you want an endgame quality weapon? Hold out for a quick blade or demon weapon then. If you just want something to tide you over through the midgame, and possibly through the endgame though it will be decidedly inferior, enchant a more mundane weapon like a sabre or a trident (in the latter case, be sure that you have Animate Dead memorized and castable).
On Going to the Abyss
Despite what you might think, banishing yourself via a distortion weapon, even while worshiping Lugonu, is a bad idea pretty much always. Yes, you can get out at will, but unwielding distortion weapons can have other effects - random teleport + confusion, or worse, yellow to red glow. The first is dangerous, but the second can malmutate you heavily - and it's not like Lugonu is any help there. Can it be used to save your life? Yes, but it's very unreliable - banishment occurs less than half of the time (1/5 or 1/3, can't remember offhand).
Saving your life is where the ability comes in, and the only reason you should voluntarily go to the Abyss, even as an AK, before you find the Abyss portals around D:20. This ability is not to be used lightly: it costs you a decent amount of max HP, in addition to a large chunk of piety. But it's pretty much a get-out-of-jail-free card, since nothing can follow you in or hit you as you self-banish. Train your Invocations skill to get the ability so that you can reliably cast it.
What to do if you get banished
This is where Lugonu's game is a little different than others. Standard procedure for most characters who get banished is to wander around until you find an exit. Lugonu characters can depart the Abyss at will, and will find the Abyssal rune much more quickly. This means that it's to their benefit to wander around until they see something they know they can't handle, at which point they gate themselves out. Taking an exit out merely saves you a bit of piety - not generally worthwhile, except in the stupid little starting Abyss.
Later On: Corruption
This is a quirky ability, to say the least, but extremely powerful if well-used. It costs a metric fuckton of piety, but what it does is replace a large amount of the level with Abyss-like terrain, complete with neutral Abyss monsters. The monsters will vanish after a while, but the terrain won't, so it can be used to sequence break on a lot of vaults (the corruption can get through rock, stone and metal, but not the "unnaturally hard" walls used to guard a few critical vaults). The monsters, being neutral, will attack everything - including you if you stand in their way. They don't seem to use ranged weapons against you, but they will turn hostile if you attack them, and the game does not ask for confirmation. So be careful around nasties like Executioners and ancient liches.
So what is this good for? Difficult branch ends: Orc, Snake, Elf, etc. It will take out a good deal of the monsters there if not all of them, and leave your path clear for entry. Obviously this isn't something you'll use more than a few times per game, but it's definitely a fun and useful thing.