After a ridiculous number of failed attempts I empathize with all the people having trouble with this one. I failed over and over and over again, and it wasn't until I read about it in several different places that I finally was successful, and I think that I now know why.
I did it on Pillar of Autumn.
First thing's first. On my successful attempts the assassination started well above the elite. I never touched the ground, and my shields did not flare. I had no damage whatsoever. And I did not do the headstomp animation. In fact my successful one did the animation that cuts his leg out from under them and ends with jabbing the knife into his throat. I came down very close behind him, almost on top of him. When I started the assassination my crosshair was on his shoulder, though it never turned red. So, none of that matters for the achievo.
I had tons upon tons of failures, including a great many where I would assassinate the elite without getting the achievement, even when I deactivated sprint pretty early. It wasn't until I finally got it that I think I figured out what I was doing wrong.
The achievement isn't bugged, glitchy, or random... it's just that the terrain is less than ideal for it. It's doable, but it seems like the outcome is pretty inconsistent. This is not due to the height of the fall. The height is sufficient to kill you. Thus using a grenade to take some damage did me no good whatsoever. I noticed that even with grenade damage whenever I did not assassinate I would either die outright or survive with zero fall damage. This is significant.
The key is that the terrain is not entirely flat down there. It slopes away a bit, especially at the nearest spot the Elite stops at, near the grass. What happens is if your forward velocity is high enough you will simply slide to a stop instead of dying.
So, here is my miniguide. This is the procedure I used on my last attempt which netted me the achievement in about 10 minutes of trying after hours of trying other times.
First, skip the cinema by pressing A then Y.
Immediately push forward on the left stick.
Once you start moving activate sprint.
Jump over or on top of the rock and pass Emile to get to the cliff first.
Optional - Switch to the pistol if you prefer it. I found it less obtrusive than the AR in my view.
As you come to the cliff don't stop at all.
Strafe just a bit around the big rock on the right to line yourself up. To the left of the big rock are two flat medium rocks, then to the left of them are many little rocks. Forward towards the edge of the cliff where it comes to something of a point are two identical pairs of tiny rocks. I used these like runway lights strafing until I was lined up then I would run strait through.
Begin sprinting early enough to get up to full speed.
Look almost but not quite all the way down and wait until the last moment to jump. You can jump later than it seems you would be able to because the edge is a little rounded. You might lose a little altitude, but you need the distance you gain. There is plenty of altitude.
(Here you will have a decision to make. On the first run you don't even need to attempt the assassination, just setup your run and see where the elite is once you jump. If he is in a bad position or moves before you hit the ground then pause the game and restart the campaign. It only takes a few seconds to setup your run over again, but it pays off because if you setup the run quickly enough then your checkpoint save will be just before you jump off the cliff and the Elite will always be in the same position from that save. You can try as many times as you like with exactly the same circumstances that way and do each attempt very quickly as long as you don't die too many times, but even if you do just setup your jump the same way again.)
Pause mid jump. If the Elite is in a position you like and behaving how you want then restart from the last save. If not then restart the campaign until he is.
Once you have a checkpoint save you like simply jump off the cliff the same way each time, but adjust the direction you jump off the cliff based on where you know the grunt is from previous jumps. If that proves difficult then run to the edge and look, then backup and try to note the closest edge of the cliff to him. Then restart from save and see if you can make the jump at the right angle. (The more accurate your initial jump is the less correction you will have to do mid air. This means you will be able to turn off sprint sooner and freefall without adding forward momentum sooner as well. Both of which can cause achievement failure if done too late.) I think the best position the Elite can be is the one on the right, though it's a further jump than the middle position next to the grass. The left position is doable, but I had terrible luck with him usually running up the road out of range mid jump from that one so I trashed it anytime he was there or up the road.
Keep looking down and make any corrections you need to as early as possible. I found it easier to make my initial horizontal correction by aiming left or right and pushing forward on the left stick rather than attempting to air strafe, but use whatever works for you. Looking down while doing this seems to help, though I couldn't say if it actually adds velocity to your fall or not, but it definitely lets you see your horizontal velocity clearly and helps you know how much you need and when to cut it down.
Keep sprint on until you feel like you have enough momentum to make the distance to the Elite you are aiming for. I kept it on what seemed like a full 1/3 or the jump.
Once you are heading horizontally directly at the Elite aim your crosshair directly at him and track him with it. Keep sprinting and pushing forward, and use a little strafe on the left stick to make minor corrections. Don't turn off sprint until you have almost enough velocity that you could turn it off and put the controller down but still land right behind him.
Once sprint is off you can make a few more corrections, but very soon you need to stop pressing forward at all. In fact if you have too much forward velocity you may need to even pull back on the left stick. The objective here is to be falling as close to strait down as possible by the time you would hit the ground. It may help to aim to land initially to land on top of his head while you're still sprinting then once you turn it off aim for just behind his back. I would estimate that the last half of the fall you should not add any forward momentum at all on the left stick. You shouldn't even touch it except to maybe pull back on it or make small horizontal corrections, but really you should be perfectly lined up by then and have no need for it or you very well may fail anyway.
Once you are closeish to the Elite (maybe 2 to 1 times his own hight above his head) hold down the melee attack button down to start the assassination. (Be sure to be aiming at him by now at least, though you should have been already.) The timing can be tricky, but isn't that difficult, and you do not have to wait until the last possible moment. Keep in mind that the further behind him you fall the trickier the timing will become because you will have less time in range of the assassination before you hit the ground. On my succesful attempt I was so close I might have actually bumped him before the ground if I didn't assassinate him.
Really the key to the whole thing is the proper jump off of the cliffside and the first 1/3 of the jump with sprint on. Just try over and over. If you've setup the save like I said then you can cram in jump after jump after jump. Just be ready to pause halfway through the jump if you aren't lined up. If your jump isn't perfect by halfway then you'll probably fail to get the achievement anyway even if you do manhandle yourself into position for an assassination. So go ahead and restart from the checkpoint. That way you won't die trying as many times. Remember to pause and restart as soon as you deem a jump a failure. And anytime you get an assassination without getting the achievo restart before the grunts kill you. (Not really a huge threat unless they use a grenade, but they tend to sometimes.) Try to prevent yourself from dying because once you die too many times you'll have to setup the check point again from the start of the campaign. As long as you don't die you can retry over and over again with only seconds per attempt. In fact the sooner your pause and restart the faster it goes. So abuse it. Pause and restart as soon as you jump off the cliff until you are jumping off at just the right angle. Then let yourself fall a little further and restart at or before the 1/2 mark until you think you've nailed the jump. It's just like the movie Groundhog's Day, only compressed into mere seconds.
Finally, somebody made a suggestion that I think can be good training. Try doing the jump several times, but without actually using melee at all. Just try to make sure that you consistently die doing the jump exactly the same way each time. If you survive any attempts then you are doing something wrong and need to come down with less forward momentum at the end.
I've uploaded to file share a short movie showing me setting up my check point and 2 failed jumps before my finale successful one if anyone wants to see it. You can just lookup by my gamertag (BADmannon).
Also, don't lose hope. Keep trying. In this case it is very very possible to get the assassination without meeting all the achievement requirements. Failed attempts can look very similar to successful ones. Even doing everything pretty much right it may take many tries. Just strive to get closer and closer to the perfect jump where you stop sprinting or using left stick 1/3 of the way down, fall strait down 2 inches behind an elite with your crosshair right on his head, and die every single time if you don't assassinate him. It doesn't actually have to be perfect, but if you strive for perfection then you will be pleasantly surprised when you actually do get the achievo.
PS. Sorry this is so long, but it was getting really frustrating for me until I finally figured out what was going wrong and why and I've seen a lot of other people with similar problems. Simply knowing what worked for others wasn't enough for me, I had to figure out why some things worked and actually understand the problem before I could fix it. So this is for people like me that just need more info.
PPS. If you setup your jump as I suggest by simply sprinting ahead and jumping off the cliff ASAP when the campaign starts you will mercifully cut off a lot of the chatter and only have to listen to a few words of it over and over each time you jump and restart from last save. ;3 In fact once I started doing this I think it actually helped me with my timing and no longer really registered as a sentence, but just sounds because I was focused on the jump and he really only said a couple of words each time. A nice fringe benefit... Also since you're in front of him he never gets in your way.