Livelock

Most of your kills will come while working on all the other achievements first. Once you have done everything else you can grind the rest of your kills out on 1-8 Last Stand on easy.

I found 1-8 to be the fastest, because all you do is stand in a small arena and kill enemies. I used Vanguard because his seismic wave attack could be used to kill enemies on one side, while attacking enemies on another, but any chassis will work fine.

Once you clear all five waves, and the objective is to enter the pod, just hit restart mission to avoid the score board.

Using this method I would get around during 200-250 kills in a few minutes. Hope this helps anyone that needs to grind this out like I did.

As Jduke705 has pointed out, doing this solo will be the most effective method. Kills do not share when playing in co-op.

After finishing the campain on hard and collecting all audio logs, you'll probably have only have to grind kills and level the other chassis (classes). If you only have one of those ramaining, please follow the specific guides

I found that Mission 2-1 is the most effective level to combine both.
You will get 150-200 kills and 15,500 XP in about 2.5 minutes by following the steps below:

1) Start Mission 2-1 on easy
2) Head to the mission pointer (presss to run) and kill some smaller enemies while passing them.
3) On your way you will meet Garris (huge guy) for the first time. Don't kill him, but shoot him for 1-2 seconds (See notes at the end!)
4) Head further north-north-east and enter the barn you will see (glowing entry).
5) Press the button in the back of the barn. This will trigger 130 mini spiders, which will just drop in front of you and can be destroyed within seconds.
6) After the spiders stopped dropping, leave the barn and continue to the mission marker. Don't stop for enemies, but kill the bugs on your way
7) Collect the mission objective and leave the level through the exit further east.

Notes:
- When heading straight to the barn without killing some smaller enemies and/or damaging Garris, the spiders won't spawn or very slowly
-Skip all conversations by pressing X!!! Otherwise you can't collect the objective and leave the level until the conversations are over

The other two guides are brilliant, however I found that the best efficiency was to do the following:
Character: Vanguard (Ensure Seismic Wave is equipped).
Level:1-8 Last Stand.
Method:
Wave 1: When starting, head down the right hand path where the enemies spawn and spam hit them till they're gone, this will net you around 50 kills and 0.5% of the overall achievement. Once this is done, go back to the middle and finish off any stragglers.
Wave 2: Clear the 4-5 big enemies that spawn next to the right turret, lots of smaller enemies will then break the ground and come from under a building, kill all of these quickly and it should net you around 25 kills and 0.25% of the overall achievement. Once this is done, head up and perform Seismic Wave ability on all the enemies grouped there for another 30-35 kills and another up to 0.35% of the achievement. You can then continue upwards and finish off the smaller enemies that spawn in to net another 5-10 kills. and up to 0.10% of the achievement.
Restart mission and repeat both waves.

With this method you will earn up to 1.2% of the achievement in around 90-120 seconds.

Tips: Skip through cutscenes and speech using "X" to speed up a few more seconds.