Important: Liquid Metal Slimes and Metal King Slimes do not count towards this trophy.
In order to obtain this trophy you will need to specifically kill a Metal Slime. The best location for this is in "Arba Town - Recruit". Eventually while fighting a random pack of slimes will appear on the map. Your main character will also need to land the killing blow on the slime, so do not be confused if it appears in your Monsters Battle Records as an AI may have landed the killing blow. In order to guarantee this, solo "Arba Town - Recruit".
For fist time slime chasers, in order to easily get close to the slime, save your tension meter for later and activate it when you get close to one allowing you to run faster. Do not attack the slime or activate your tension meter until you get close and are ready to battle.
It may also help to have the "Magnetic Necklace" accessory equipped to increase the spawn rate of metal monsters.