Right, so basically, it's not as complicated as it sounds. To acquire the achievement all you have to do is complete round 25 on any Exo Survival map. When this occurs, you'll learn that the map will now be "flipped" and the round number will revert back to 1 (but 12), to start again.
I'd suggest that you play in a 2+ session (I'm clueless as to how solo pans out) and choose different classes - so one would be Specialist whilst the other's Heavy. This way, you'll gain different scorestreaks which will help you to get by. The enemies will get harder (and more tactical) after the flip, but that doesn't really matter as you'll have the achievement! Yay.
Tips:
-The map I used was "Retreat", playing Co-Op. Me & my friend camped (...I'm sorry) on top of the stairwell that is positioned inside the building next to the Armory, in a "back-to-back" fashion, ensuring we didn't become the victims of flanking. Note that enemies also possess Exo abilities and can therefore jump up to you, so be careful. When it came to the Objective rounds (that cycle between defusing bombs, collecting tags, retrieving intel from fallen foes & getting 'defend' kills), my friend would kill them off whilst I was completing the objective, so I had protection. Also note that if you fail an objective, you're going to come across some form of setback - which, I've seen, can be pistols only, enemy smoke grenades, a system hack & nano strikes - so I'd suggest you make an effort to complete the objectives.
-Goliaths/LMGs are useful when "Mechs", a.k.a Goliaths, arrive. They can also be taken out with 4 melee attacks (using your knife) so if there are two or more players, just gang up on the poor soul. I'm unsure if the amount of melee attacks needed changes but it's 4 for quite a bit.
-Pick a weapon and stick with it! The points you would otherwise use on buying new weapons could be spent on something more beneficial, like the damage/armor upgrades and also upgrading certain scorestreaks (UAV FTW).
Good luck, and I hope you succeed.