Mass Effect Legendary Edition

For most people, including myself, your kill counter will reset if you swap games. Like, if you have 1999 kills in ME1 and you click on ME2, you can very likely lose all progress (This could be fixed now for all I know, but I couldn't guarantee that)

My personal favorite method for this achievement is one you can do very right after you gain control of the Normandy in Mass Effect 1.

It involves the mission UNC: Lost Module with the monkeys you can use to glitch infinite paragon points. This mission is located in the Attican Beta system - Hercules cluster - planet Eletania.

Look at the map and go to the orange marker that is on the right side of the map. You will start the mission here.

The correct area out of the three monkey colonies that are highlighted on your map is the middle one. It has a structure next to it (you need to go in here to finish the mission).
Drive up the structure and park in between the monkey nests. Save your game.

Use the Mako to kill all monkeys, reload, and repeat.

I was killing 10 monkeys about every 10 seconds. Starting from 0 kills (because of a glitch), I reached 2000 kills in about 30 minutes.

Edit: Per Fiezza below, powers may not contribute to getting kills, so it is wise to do gun only runs to make sure every kill you get is counting.

Edit 2: I forgot to mention that I was using the M-490 Blackstorm Singularity Projector for my first few runs and I was only seeing an increase of 3 kills per run even though it was wiping out every enemy. Best of stay away from that gun specifically as well and maybe all heavy weapons.

This method will generate between 40-60 kills per run depending on your luck/weapon levels. Each run takes anywhere from 2-4 min. I was able to grind out the last 1000 kills I needed in about 1 hr using this method. I was on the casual difficulty to make enemies as easy as possible. This method uses ME2 since that is currently the only reliable game that tracks, even with the update.

The mission you are looking to use is "N7-Abandoned Mine" which is found on the planet Aequitas (Minos Wasteland, Fortis). Credit to Neo Yggdrasill for his comment in the forums pointing me in this direction.

I recommend a save before you go to the mission in your cabin on the Normandy for an easy load to check your progress.

After that is done, land on the planet and make another save incase you mess up and end the mission. Now progress through the mission until you come to a path that branches off in 2 directions. Make a separate save here.

From this point, move up to the right and onto the overhang, notice the crate on the right that has an ammo resupply, do not pick this up yet. Once you enter the overhang, enemies will spawn. Go to the top of the ramp but do not go down to the bottom. simply let the husks run to this ramp and up at you. Aim for their torso for an easy 1 burst kill with the assault rifle. Use the ammo on the crate as required to finish out the wave. You will get an auto save once all enemies are defeated.

Once complete, head down and collect ammo as needed, there is always one right at the bottom of the ramp. Then head in the doorway to the right and continue into the room ahead. Once you have cleared the doorway, quickly pop back out and look to your left. There will be waves of 3 husks that spawn for a total of 6-7 times. Simply keep killing these until they no longer spawn. you will notice another auto save happening at the bottom.

Reload your save from the bottom of the ramp earlier and repeat as needed.

Heads up the tracker resets if you return to Mass Effect 1 after starting Mass Effect 2. I was about 100 short by end of 2 loaded up 1 to mop up some others and the stat checker in 2 is back at 0 kills.