This is the long haul achievement for coop. You will definitely hit rank 30 and probably have the majority of your upgrades (52 for apps, 86 for weapons) before you're even close to finishing all of these. Any enemy that must be breached to be killed (Liquids, etc) can be shared between two people so long as you breach the final armor layer together, so you can easily cut down those grind times. Here's the list of what you'll need to accomplish for the challenges:
Total Kills: 5000
Includes all kills (even liquids, etc)
Spectre Kills: 50
Found in Cayman Global maps (1-3)
Electron Mace Kills: 100
This only counts people you kill that are holding the weapon. Sergeants in Colorado, China or Scandinavia.
Riot Lance Kills: 100
This only counts people you kill that are holding the weapon. Police in China or Scandinavia.
Riot Shield Kills: 100
This only counts killing police that are holding one. They show up in China and Scandinavia.
EMC Kills: 50
They are the guys that jam your DART chip and breaching abilities with the pulsing animation. EuroCorp levels (7-9).
Aspari Kills: 1000
All Aspari kills count to this on any Aspari themed level (4-6).
EuroCorp Kills: 1000
All EuroCorp kills count to this one on any EuroCorp themed level (7-9).
Cayman Global Kills: 1000
All Cayman Global kills count to this one on any Cayman Global themed level (1-3).
Police Kills: 100
These guys only show up in China and Scandinavia. They wear black attire, so you can tell them apart from the gred/red Aspari guys.
Mercenary Kills: 20
They will randomly spawn on Expert mode on a few levels. Western Europe, Colorado and Scandinavia are all confirmed places where they will show up. In Western Europe, they will drop near the first giant door. In Colorado, they will drop on the landing pad above where you start toward the end of the level. In Scandinavia, they will drop after you leave the warehouse. There's no sure-fire way to get these guys to show up, but when they do, around 7 of them will show up. This will be the most difficult challenge to solo, so grab a friend or three.
Liquid Armor Kills: 50
They show up in Cayman Global levels, but will be replaced in higher difficulties. Hard is probably the best balance of them. You might have to farm these, due to how uncommon they tend to be compared to reactives.
Electric Armor Kills: 50
These are EuroCorp faction units, so they show up in levels 7-9. They are functionally like Liquids, but emit electricity when ready for a chip rip.
Reactive Armor Kills: 50
They are in most levels, and are usually replacements for liquids/electrics on higher difficulty levels. They are more plentiful in the long run than either Liquids or Electrics, and you will encounter them on every level with Scandinavia probably having the least.
Agent Kills: 20
These are the enemies that usually show up at the end of the themed arcs. The following levels have agents:
4 Agents at the end
1 Agent in tunnels (hard difficulty or above)
4 Agents at the end
2 Agents at the end
4 Agents at the end
Kills by Fire: 100
Thermite thrower, COIL, and CQC shotgun's alt-fire will accomplish this. Since CQC is a loadout gun, it's easier to do it with that since you can reload it at any ammo box, and it has a decent range.
Gibbed Enemies: 500
A gibbed enemy is an enemy that dies when they lose an arm, leg, are split in half, or lose their head. Shotguns at close range can accomplish this, as can the sniper rifle, Bullhammer revolver, minigun and explosive weapons. The sniper is a good choice for this, because you can shoot heads from behind cover and also go for penetration kills at the same time.
Melee kills using the right stick. This also includes chip rips.
Healing your ally for any amount of health counts. C2C heal is a great way to pump this up, otherwise, normal heals and squad heals count as well.
When your team mate is down, they will call out for a reboot. Breaching them will raise them again. Do this 200 times.
This is awarded when you heal someone that's about to die, or kill someone that is using an electron mace on your ally.
Team Savior: 50
Only awarded when you reboot someone as the last man standing. Doesn't matter how many other players are in the game. Team Savior only counts for the first reboot when one person is left, so raising your partner in a 2 player game counts the same amount as raising one partner in a 4 player game.
Syndicate Heals: 500
It's the same as the "heals" challenge, only for healing a member of your syndicate. This stacks with normal heals, so you can complete both challenges by healing people in your syndicate 500 times.
Syndicate Reboots: 200
Same as reboots, but when rebooting a member of your syndicate. This stacks with the "reboots" challenge, so you can finish both challenges by rebooting a syndicate member 200 times.
Syndicate Savior: 50
Same as savior, but when "saving" a member of your syndicate. This stacks with the "savior" challenge, so you can finish both challenges by "saving" a syndicate member 50 times.