The Mechs you have to collect are:
Light: Locust, Commando, Spider, UrbanMech, Firestarter, Jenner, Panther
Medium: Cicada, Blackjack, Vindicator, Centurion, Enforcer, Hunchback, Trebuchet, Griffin, Kintaro, Shadow Hawk, Wolverine
Heavy: Dragon, Quickdraw, Catapult, JagerMech, Thunderbolt, Cataphract, Grasshopper, Black Knight, Orion
Assault: Awesome, Victor, Zeus, Battlemaster, Stalker, Highlander, Banshee, Atlas, King Crab
All you need to have is any one copy of each Mech, as opposed to one of each variant (which would involve hunting down 59 variants, and would suck). You must have every Mech in your possession, either active in the Mech Bay or in storage, at once for this to pop. You start the game with a Blackjack, Shadow Hawk, Vindicator, Spider, and Locust, and will be granted a Centurion. As far as acquisition goes, remember that enemy Mech tonnage increases based on milestone Priority Missions. This does mean that you'll need to play the Restoration campaign to gain access to every chassis, and will probably need to do a fair bit of grinding unless you're godly at beheading Mechs. For easiest collecting, I'd recommend you make sure you've collected every Light Mech before starting First Strike (except the UrbanMech, which is only found in stores as of v1.0), and every Medium Mech before Liberation: Smithon.
This achievement seems to currently be buggy for campaigns started before the 1.0.3 patch. A potential fix seems to be readying all Mechs that were in storage before the 1.0.3 patch and returning them to storage. You may also need to salvage/purchase another Mech to trigger the achievement.