After you return to the Sector 5 Slums from Aerith’s House during Chapter 8 (after picking flowers) you’ll have to rescue some kids. Do so and you’ll be able to start some side quests. Complete Kids on Patrol to unlock A Verified Hero, the object of which is to compete in their newly-devised “Whack-a-Box” game, the goal of which is to destroy a variety of boxes as quickly as possible. The boxes you’ll find stacked about have different point values, and your goal is to score as many points as possible within the time limit.
With a little practice, it should be easy to hit the 30,000 point mark - your goal for this trophy. Equip the Iron Blade and use “Triple Slash” every time you have the ATB to do so. Aim for the Time Boxes to extend your time, and focus on stacks of smaller boxes, as they’ll ultimately get you more points faster than the large boxes will, which are rather sturdy. Check out the page for Whack-a-Box for more details.