This achievement can be difficult but not impossible. What you need first and foremost is four sensible, great players who are either high level (>22) or prestiged, who are good communicators and will work in a group.
Most of the challenges are fairly self explanatory and easy enough to obtain when working together. Some require you to put your turrets away (face them against a wall), such as melee killing or killing with a certain class of weapon. It is best to not setup new turrets until you know what the challenge actually is (after the drill has been set). Here's some of the tricky ones:
* Don't reload for the first 60 seconds: Be aware that throwing out ammo and sometimes switching weapons can trigger an automatic reload.
* Maintain an accuracy of 50%/75% or above: determine which person in the group is the best shooter, and let them make the necessary shots. It's best to perform this when an alien is attacking the drill. Turret bullets luckily do not count, so go nuts with those and rely on melee for defense or cleaning up the mess. Keep your finger off RT!
* Kill the leper in the first 30/60 seconds: depending on which wave this comes up, this can really hard to get. However as per usual use a flare and fire on anything that moves - 90% of the time this should be good enough. As started in other solutions, KILLING THE LEPER WITH A TURRET DOES NOT COUNT, so tuck your turrets away temporarily. (Note: there are reports that this challenge may have a 30 second limit now.)
* Kill 5 aliens while they are in the air: electric fences are your best bet, otherwise try to get up high like on a fence, roof, top of truck etc and time your shots.
* Kill 25 aliens with a sniper rifle: Abandon all hope. If you want to give it a try, grab a sniper rifle AND then grab new ammo (armor piercing hopefully, explosive as a fallback) as your previous ammo doesn't come with you. Use a flare to draw aliens into one area and fire away. Preferably get a number of people in your team to get the sniper rifle. I haven't seen a team ever get this challenge, but I'm sure it can be done.
If you're playing to grind, time efficiency would say that if you fail a challenge before the second barrier hive you should end the game and start over. If you fail a challenge after that, it's still worthwhile doing the rest of the waves for practice.