Challenges are not available immediately when starting the game, and for a few levels after you move up a generation. When they are, you can choose Challenges in the menu and look through the various weapon challenges to track your progress. This achievement requires that you complete all available level V challenges for a single weapon. Note that challenges reset each time you move up a generation, so this has to be done in a single generation. Depending upon your play style, you may do this naturally as you play.
Play Last Titan Standing with one or two opponents, and use the Mag Launcher as your Pilot's anti-Titan weapon. At the start of the match, exit your Titan (hold down for a few seconds) and put it in Guard mode by pressing once. Cross the map on foot and destroy the enemy Titans with your Mag Launcher. After your tenth kill, the level V challenge will complete and the achievement will unlock in-game.
You can choose to focus on an anti-Titan weapon such as the Mag Launcher (which is the most survivable anti-Titan weapon to use since it can fire from outside the line of sight of an enemy Titan) or any anti-Pilot weapon you are comfortable with. Use the menu to keep track of what you need to be focused on for that particular weapon, and then just keep playing matches until it unlocks. The Smart Pistol and the R-101C Carbine are both easy weapons to use and the challenges are straight forward and simple, if somewhat longer than the anti-Titan weapon challenges. Challenges related to Spectre kills will progress much faster in Frontier Defense games than any other mode, although Sniper Spectres and Suicide Spectres do not count for challenges.