After you complete the quest Land of Convergence you'll transition into post-game play, which largely centers around hunting Tempered monsters. A Tempered monster is essentially the same as its High Rank variant, with the same health and same material drops, the only difference being that a Tempered monster will deal more damage than its non-Tempered counterpart. The other differences - namely the Streamstones they drop - are irrelevant for obtaining this trophy/achievement. What is relevant, however, is how you go about hunting Tempered monsters, as they are mostly encountered via investigations (aside from a few odd assignments, optional quests, and events). To unlock these investigations, you'll need to go on High Rank expeidtions (ideally after the aforementioned Land of Convergence assignment) and find blue monster tracks. Search these monster tracks and you have a chance to unlock an investigation to hunt a Tempered monster.
It should be noted that Tempered monsters are divided into three tiers, or Threat Levels. Threat Level one monsters (Anjanath, Barroth, Jyuratodus, Paolumu, Pukei-Pukei, Radobaan, Rathian and Tobi-Kadachi) can be hunted when your Hunter Rank is below thirty, while the other two tiers require higher Hunter Ranks and the completion of an assignment... the completion of which will unlock your Hunter Rank advancement again until the next assignment. When you hit Hunter Rank twenty-nine, you can take on the assignment Beyond the Blasting Scales, which unlocks your Hunter Rank up to fourty-nine and allows you to tackle Threat Level two Tempered monsters (A. Rathalos, Bazelgeuse, Black Diablos, Diablos, Lavasoith, Legiana, Odogaron, Pink Rathian, Rathalos and Uragaan). Finally, at Hunter Rank fourty-nine you'll be given the assignment Thunderous Rumble in the Highlands, after which you'll hit Hunter Rank fifty, allowing you to hunt Threat level three Tempered monsters - namely Elder Dragons (Kirin, Kushala Daora, Nergigante, Teostra and Vaal Hazak).
Kill any fifty Tempered monsters - something you'll likely need to do to boost your Hunter Rank for trophy/achievement "Established Hunter" - and you'll pop this trophy/achievement.