As you make your way through Greenpath, talk to the Hunter and he will give you the Hunter's Journal:
You can access the journal from the menus by pressing
. Each time you kill an enemy, its entry is added to the journal. Once you kill an enemy a predetermined amount of times, you unlock the Hunter's notes on that enemy, For this achievement, you simply need to kill each enemy one time so it is added to the journal, but in order to receive the Hunter's Mark for the True Hunter achievement you must unlock all of the Hunter's notes.
There are numerous enemies that are not actually required for this achievement, likely because they were added as DLC in the PC version of the game. The following images shows the 150 entries I had when the achievement popped on my alt account that I used when I wrote the site walkthrough:
Compare your journal to mine to see what you're missing. If you need help finding specific enemies, look them up in the Hollow Knight wiki