So, for this rather annoying achievement, here's some suggestions.
First, basic mechanics:
-In this training mode ONLY direct impacting damage, such as bullets and melee (ie: splash damage will do nothing), will trigger a target, making some guns and most grenades plain ol' useless (This makes sense, as it blocks things like the Jericho, even if it's disappointing).
-You probably want to stick to the Normal Mode (15 targets in 20 seconds) instead of Challenge Mode (20 targets in 25 seconds, with additional Claptrap targets causing an auto-fail and additional Jack targets that auto-destroy all non-Claptrap targets currently active for you)
-You can do this challenge from anywhere around or inside the target arena, you don't have to stand behind the counter.
Solo: If you're going for this solo, probably get yourself a Dictator assault rifle. When doing this, keep it in the alternate firing mode, Bipod. This increases the bullet count from 3 to 7, and in a flat horizontal spread! It fires quite fast, making it so it's less about aiming in all directions and more about just up/down. Additionally, when in the Bipod mode, if you look down the scope it tightens the spread, making it easier to hit the further targets.
I got this suggestion from the guide posted by Recster on TrueAchievements, then finally unlocked it a few tries later!
Co-op: Well, I'd probably still suggest using Dictators, but the main thing is that you can divide the work between the players (only the person who pushes the start button will get the achievement, so be sure to take turns!). Though, if you have one person handling all the grounded targets, I'd very much recommend them using either a Boring Gun (since it fires out multiple rebounding sawblades; even better with the Double-Penetrating prefix), ) or a Fingerbiter (which fires out many bullets that ricochet up to four times each, so just fill the room with them! Mainly only good for the lower targets, due to the parabolic arc and slower travel speed).