On the main menu, select Arcade. Once you're in Arcade mode, press
and select your preferred Arcade Hero mode. I got this achievement by selecting "Arcade Hero - Solo and Co-Op" and was randomly transported (solo) to a very easy map that allowed me to hijack a helicopter and kill a target that put up no resistance. I unlocked this achievement in a few minutes after playing this map a few times (upon completion, you can replay the map that was randomly selected for you if you want).
I can't say for sure whether killing the target and completing the mission mattered for the achievement, but based on the achievement description, you can probably just play any map to completion, win or lose, and make progress towards this achievement.
I'm also not sure if there is some way to select specific maps in this mode. After getting the achievement, I tried this mode out several times and never found a way to select a specific map (you can, however, play another random map by pausing the game and selecting the "play another map" option, which could land you upon an easier map if wins are needed).
If anyone can confirm that losses count, or if there is a way to select specific maps, please let me know and I'll update the solution.
Edit: According to pezboyben, losses do count towards this achievement. Just play any 5 Arcade Hero games to completion and you'll get the achievement.