On stage 7 "Some kind of Lab" you begin in a room with a punching bag you must hit to continue. Keep hitting it until you hear a chime noise then stop.
You will be teleported to a new area with quite alot of enemies followed by a platforming section. Upon completion of this area the achievement will unlock.
Note: Try to stop hitting as soon as you hear the chime. I'm not syure if it works if you keep punching past it before stopping.
Credit to 'Psycho0022' for suggesting it unlocked for him at 87 hits, which may be the intended number of hits required as 1987 is the year the first Double Dragon came out.
Credit to 'celestial purge' confirms that if you equip successive strikes you get a counter for how many hits you have done, and says for them, it unlocked at 97 hits
Credit to several comment posters and myself included have confirmed that if you keep destroying the missiles after about 5 minutes they will stop