DOOM

For this achievement to unlock you have to equip an Cyberdemonic armor piece in multiplayer.
The fastest way to earn this is to level up to Epsilon 8 and play a minimum of 5 more matches until you complete a challenge. After you've completed the challenge, equip your armor piece and play one final match.

The quickest challenges are:

MVP? More like LVP 1
Kill the top enemy three times in a match.
(5 times)

This one will unlock the TORSO armor piece: CD1: RUINER

Bulletproof 1
get 5 Hard to Kill (Best kill/death ratio) commendations.

This one will unlock the TORSO armor piece: CD4: HELLBURST

It took me 16,5 hours to get this achievement, all done within the double XP weekend and the bounty rune.

This achievement will be unlocked after 12 hours of efficient Sector boosting, DURING the 2x XP weekends. If you go for it any other time when it is standard 1x XP, then it will take 24 hours of grinding to obtain. There are a couple of steps involved to obtain the Cyber-Demonic armor. You need to get to Elechon 8 and then complete a 15 kill / 0 death streak. Once you complete that simple challenge you will be rewarded with the armor. Equip the piece of armor during the next match and the cheevo will be yours.

Boosting in a Sector match is the most efficient way. It is KotH, with 2 hills. You can manipulate where the 2 hills are in general, so ideally you move them as close as possible. In this mode you will NOT be killing, just running back and forth. The video below shows the 2 hills right beside each other. Everybody stays to the right and just keep going back and forth for easy XP.

Thanks to BLACKPANTHERv88 for the video he provided me as an example

The initial game time is 10 minutes. As long as the score is tied as the 10 minute timer expires, you will proceed through Overtime which is 2 minutes in length. As long as the score is still tied after the overtime period, you will then go into Double Overtime, and then Final Overtime. You can have up to 16 minutes each game to boost, if done properly. You can get upwards to 100,000K points per game, but will most likely average 50-70K per match.

You will rank up 1 level with every 10-11 captures. (thanks DURDENS WRATH for pointing this out in game)

There are 3 challenges you can do in game that will net you 2K each and involve NO killing:
1st one is to equip the chaingun and keep it spun up. You just run around with the button held for a match.
2nd, is Threat Pulse. You send a sonic pulse with to mark the enemy for a few seconds.
3rd, is Threat Detector (?). You drop a mine that pings with an enemy is detected by it.

When you set up your session, make sure you select the ASIA server, for it has zero traffic.

I took this screenshot from the last session I was in.

As of the 6.66 update, the way the former-DLC works is slightly different. You either have to earn enough XP to unlock the item(s). or complete a challenge to unlock them. Either way, armor pieces will be locked until you hit a certain level.

The earliest you can unlock a single piece of Cyber-Demonic armor as of this writing are a few challenges unlocked at Echelon (DOOMs version of Prestige) 8. Without challenges, it looks like the first piece unlocks at level 13, Echelon 10.

So if you didn't get it from the DLC before it went free, you're in for the long haul.

QUICK EDIT - You actually unlock the first piece once you hit Echelon 10.

To add to the other excellent solutions for this achievement, another easy challenge to get a Cyber Demonic piece is after getting to Echelon 8 is to get the World Destroyer I challenge which is to get a 15-0 killstreak which unlocks the TORSO CD: 3 Deathgrip

A long grind, no two ways about it. You need to reach Echelon VIII level 1 before the milestones for the first torso piece of cyberdemonic armour become available. The easiest challenge we found for this was the Mayhem medal, which you get for a 15 kill streak. You have to start a new match after reaching Echelon VIII to be able to unlock this. Once you unlock the armour piece, the achivement will unlock once you start a match wearing it.

The fastest method (deveoped with @JayD101 who was very kindly boosting this with me) was to play Sector matches. The Asia Pacific servers are usually empty, however if you have people who keep joining your matchmaking, blocking them from their xbox profile and then restarting your search clears them out nicely.

You need a minimum of 4 people to start a game in the team play list. Private matches unfortunately do not grant xp or achievement progress.

The match is random each time you search, but you can back out and search again until you have a sector match. Have all players do this (preferably in 2 in-game parties of 2) and you should find each other relatively quickly.

Have both teams capture and extract a sector at the start of the match.

Then have the booster and a player from the opposite team take turns capturing the same zone over and over for the rest of the match. The match will go into overtime, double overtime, and then final overtime, granting you 16 minutes of boosting. In the final 30 seconds, let the booster extract the sector to get the win for an xp boost.

To massively speed up the process, have the 2nd player on the opposite team run up to the booster repeatedly during this, allowing them to be killed. Try to kill them with different weapons over the grind, or use melee and glory kills. There are a lot of different medals and challenges and you can earn 2000xp for the higher levels of these, which is a significant bump.

Doing this, we were able to earn 10 levels a match over 16 minutes, or an echelon every 5 matches. If anyone joined mid game, we would block them and then restart the match.

Every level takes 5,000xp so your level up progress is pretty consistent.