Dead Cells

Conjunctivius is found by traveling through the prison -> Toxic Sewers -> Ancient Sewers - Insufferable Crypt. You will need the Ram and Vine rune to access.

The above solutions are great, but I found an easy strategy was to use the heavy crossbow, wolf traps, and double cross-bo-matic. If you place the traps on the first platform, you can bait him into getting stuck in the wolf traps. Assuming you have collected enough points in tactics and have decent level gear, you can kill him without him performing any attacks. This one really isnt too bad - ratio is really deceiving.

Short 30 second clip below demonstrating.

Conjunctivius is the eye-monster in the Insufferable Crypt, found by using the Ram Rune to take the lower path in the Toxic Sewers, through the Ancient Sewers.

There are two main components to the fight, which you'll want to practice several times before trying for a perfect run. Conjunctivius largely floats around the chamber attacking you, but at approximately 90% health, 70% health, and 40% health he will retreat behind a shield and summon tentacles to attack you from under the ground.

Floating phases: It will move around the chamber, periodically making these attacks.
* A ramming attack telegraphed with a ghostly red chain.
* Shooting equally-spaced green spit projectiles (more bursts at lower health).
* A close-range area-of-effect burst, telegraphed by a red circle.
* At half health or less, a long barrage of chaotic green spit projectiles reminiscent of a bullet hell. You should be doing enough damage that Conjunctivius doesn't get a chance to do this.

Tentacle phases: These phases, especially the third one, are the hard part of unlocking Flawless victory. I'll describe the tentacle behavior here.
* There are three tentacles in the first tentacle phase, four in the second, and five in the third.
* Each time you kill a tentacle, lightning flashes onto the other tentacles. The lightning doesn't hurt you; this is a brief respite.
* In general, tentacles will be submerged in the ground and follow your movement. It's unclear, but I do not think they can take damage over time while they are submerged.
* Tentacles will burst out to attack when you or a turret is over them. This is fairly easy to dodge if you keep moving.
* In the second and third phases, after the tentacles appear and after a tentacle is killed, one tentacle will make a sweeping attack that is the trickiest part of the fight. It will move, submerged, all the way to one side of the room, and then emerge and sweep across the room before submerging again and moving to your location. This attack is also easy to dodge in isolation, but in combination with avoiding the other submerged tentacles it will trip you up.

Strategy
If you're aiming specifically for this achievement, focus entirely on one stat to maximize your DPS. On my successful attempt, I went all-in on Tactics, using a Wolf Pack trap to get Conjunctivius trapped on the lower platform so I could unload a few Heavy Crossbow blasts into it at a time. The Wolf Pack also traps the moving tentacles, making that phase much easier. Of course, any ice attack is very useful for that too. I would not recommend relying on turrets or DoT effects because they can obscure the ground and you need to see where the tentacles are moving around under there.

Recommended gear (I'll add recommendations from the comments as well):
Skill: Wolf Trap
Weapon: Ice Bow
Weapon: Heavy Crossbow
Grenade: Ice Grenade
Mutation: Efficiency / Fireworks Technician
Mutation: Support

----------NOTE:-----------
Be aware this guide was written before a patch was done that changed the game's behaviors. I do not know the full extent of the changes, so I have not updated the guide outside of this note.

changes I do know have happened:

- assist mode no longer disables achievements. this should make getting this and any other no hit achievements much, much easier.

- supposedly the 'save state' system outlined in this guide has changed. I'm not 100% on the changes, but from what I'm told it still sort of works but you can no longer save scum your way to a no-hit victory.
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There's a trick to getting this and any other no-hit boss kill achievement.

Dead Cells has a sort of quick save/save state system. This is at any time you press , the game will save at exactly that point wherever you are. The moment you die (as in, hit 0 HP and do the death animation), this save feature is disabled so if you press , dashboard, and quit the game you will continue off the last time you pressed while alive. If you never did, you will restart at the very start of whatever level you made it to. This trick will allow you to retry for the no-hit achievement without having to restart from the beginning of the game.

note: if you get hit, you want to be completely DEAD (as in zero hit points) before pressing , or else you will create a save and void the achievement for that run.

Conjunctivitis is kind of easy as long as you haven't gone crazy unlocking weapons.

basically beat the boss down until there's that brief resting phase and he summons his tentacles. This is a great place to save.

obviously it helps tremendously if you have the +, ++, and S blacksmith upgrade chances as high as you get them.

my suggestion:

every upgrade scroll you find should be spent on TACTICS (purple lightning bolt)

Weapons:
x2 throwing knives (in general 2 throwing knives makes the whole game easier) - these are EXTREMELY GOOD against conjunctivitis since you will auto-aim your throws close to 180 degrees, so you can hit him as he moves around. update: the ability to x2 the same weapon has been removed. I have not played since this change, but:
throwing knife - this is a great weapon for conjunctivitis for the simple fact throwing knife will target enemies close to 180 degrees from your position. highly useful characteristic given this boss moves all over the place

Traps:
wolf trap
double crossbow-o-matic
(both of these are handy for the tentacles)

Mutations:
ammunition
support

I was struggling with this one for a while today until I found a way to get through the tentacles consistently.

I had a buzz saw trap, which I would plant far right of the map so I could see the ground.

Getting through the first phase is easy enough. It’s that damn sweeping tentacle that caught me every time.

Immediately starting the second phase I would dodge and attack the first tentacle since it still moved slow. After killing it I moved immediately to the right trap and climbed the wall.

Once the tentacle swept to me (and there’s usually one waiting right below you, so if you straight drop you will take a hit). I found that if I jumped to the middle of the map, followed by rolling right as I was about to hit the ground I had enough time to land 2-3 attacks then head back to my trap and up the wall.

By the time you get back the tentacle is sweeping again.

This pretty much eliminated the randomness of that tentacle that caused much pain.

After I figured this out I got the flawless my second try with this stat. I’d imagine you could take any load out with you.