EDIT: A March 2012 patch has made this achievement slightly more difficult to obtain. The simplest solution appears to be clearing your cache and using my guide to get this achievement offline. Please read the end of the guide for additional post-patch tips if you'd like to attempt to get it online so you have a date/timestamp.
This is a missable storyline achievement. When you are in the level Broken Stair, you will open a chest and find an item called a Beholder Key. You will then use this key on a blue gate that has an eye in the middle of it. Right then, stop and make a manual save, because you may need to reload the save a few times to get this achievement.
After a cutscene, you will be on the back of a dragon and on a rail sequence for about 7 minutes. You will be told the most important command--that you can hold the Right Trigger, "paint" multiple targets by moving your targeting reticle over them, and then hit multiple targets when you release the Right Trigger.
There are not a lot of enemies in this section, and you may be frustrated with a few runs of only 120-140 kills. Keep in mind you can press Back and look at your kills before, during and after the run to see how close you are to 160.
The trick to this achievement, even on Apocalyptic difficulty, is that anything you paint is still auto-targeted and killed even when it is offscreen. So when the swarm of bats flies towards you in your upper left, just paint them all by holding the right trigger and hovering over them, and then release the trigger once they're offscreen and behind you. Your shots will track them and kill them all.
In my final run, all I did was paint the enemies, and the only time it didn't work was for one swarm of bats that flew from behind me, on my bottom right, across the cavern, and outside to the left; in this one case, my shots were blocked by a stalactite instead of homing in on my targets.
You may be tempted to manually target the enemies, but when I stopped trying to be precise and just painted everything I saw, I got 182 kills in the section.
EDIT 4/22/12: Nikke84 and shus26 both confirm a recent patch has made this achievement difficult to obtain. By clearing your cache and trying this offline, this may be easier. RLG JSpartan117 and Fall of Zebes both got it on their first tries after clearing their caches.
VladimirK recommends using a rapid-fire controller post-patch.
Anthony Ajax states, "I've done about 5 runs now with being able to paint a max of only 5-7 targets at a time before having to release. With this being the case, I've just started to paint as many as I can while charging, release and start again to get the most done within these limits."