You should easily get this trophy, as ALL weapons break, and EVERYTHING you pick up in battles counts as a weapon, and you'll most likely use 200+ throughout the story due to how helpful/easy it is to use weapons. If you want to check how many you have broken so far, then go back to whatever hideout you currently have unlocked in the story and enter your room, select the last option which will be "stats", and it'll show you.
When you're in a battle with someone/some people, look around the small area you're fighting in for objects that have little red arrows above them when you're beside them, and press circle to pick them up, once you have them, just simply beat the enemies up with the weapon you've picked up until it's broken. You should easily grab this trophy by chapter 4-5 if you use weapons every chance you get like I did.
If you want to know how close a weapon is to breaking, in the top left next to the weapons name is a small number, that number is how many more hits it has left until it breaks.