One of the longest grinds in the game, behind only "Giant Crown Master" and "Miniature Crown Master". Every time you complete an assignment, quest or investigation, you'll earn Hunter Rank. For the duration of the main story, your Hunter Rank will be locked, only visibly increasing as you complete various assignments. The good news is that you're still earning experience even if your Hunter Rank isn't advancing. The bad news is that you have to completing the main questline Land of Convergence to unlock your Hunter Rank, at which point any excess experience earned will be applied, potentially leveling up your Hunter Rank many times immediately after the aforementioned quest ends. As a further complication, even after Land of Convergence, your Hunter Rank will plateau again at Hunter Rank 29, being locked there until you complete Beyond the Blasting Scales, and after that another barrier awaits at Hunter Rank 49, which requires you to complete Thunderous Rumble in the Highlands where, mercifully, your Hunter Rank will finally be fully unlocked.
So, simply put, complete lots of assignments, quests and expeditions, hunt many, many monsters (necessary for various other trophies/achievements as well), to boost your Hunter Rank. Complete the entire main storyline and the two assignments thereafter to fully unlock your Hunter Rank progression, then... grind some more! It's unlikely you'll hit Hunter Rank 100 without killing many, many Tempered Elder Dragons, so don't fret and try to rush things - like with most of this game's incredibly grindy trophies, you're best off playing normally, not worrying about it, and letting the grind for other trophies/achievements and late game materials carry you there. Yes, you can grind Great Jagras over and over again to hit Hunter Rank 100, but you're better served trying to kill two birds with one stone and going after crowns and materials. Also keep in mind that stronger monsters increase your Hunter Rank more.