This is the second of two achievements related to the Hunter's Journal, the first of which being Keen Hunter:
Refer to that achievement for a full explanation of the Hunter's Journal and what is required to record all of Hallownest's Creatures.
To unlock this achievement, you must obtain the Hunter's Mark from the Hunter in Greenpath which he will award you after you record every required enemy in the journal AND unlock all of the Hunter's notes. When you first kill an enemy they get added to the journal, and aside from single encounter bosses, enemies must be killed a specific number of times (usually somewhere between 8 and 16) in order to unlock the hunter's notes on that enemy.
Small enemies respawn when you leave and re-enter an area and all non-boss enemies resapawn when you sit at a bench. The Enemies page on the Hollow Knight wiki
links to individual pages for each enemy and tells you where they can be found. You'll likely need to grind out kills for many enemies to meet the requirement for unlocking the Hunter's notes, but I suggest working on this later than sooner as you will fulfill kill requirements through progression as long as you're diligent on killing enemies as you make your way through the world.
After you've unlocked the Keen Hunter achievement and then killed enough of each enemy to unlock all of the Hunter's notes, return to the Hunter and he will give you the Hunter's Mark, at which point this achievement will unlock.