Update 17/2/2019: Now only the original launch day monsters are required for this. The newly added monsters including and after Deviljho are not needed.
The description is misleading. Actually you need to research EVERY large monster. Because those hunters in the game can't do full research on Zorah Magdaros and Xeno'Jiiva, that's why the description says "almost".
Talk to Chief Ecologist, an old guy sitting in books, directly to your left every time you enter HQ, to update your research progress. You can also check progress from him or from your hunter's notes. This achievement will come in naturally if you managed to go for the crowns.
If you want to focus on this one, forge and equip a Hunter's Headgear α. The Scholar skill on it makes progress increase significantly faster.
You gain research progress by touching a monster's footprints that gives you 10 pts, and other bigger marks like scratches, mucus, feathers..etc, gives you 20 pts. Break a monster's part gives you 20~30 pts, 50 pts for kill and 90 for capture.
The cute part is, as mentioned above, you get only 50 or 90 from fighting, but you can get way much more pts by touching all marks. So if you want to do this quickly, the fastest way is wear the Ghillie Mantle and just tail behind a monster, walk around, touch everything it lefts. It doesn't matter you kill it or not. It's possible to gain full research of a monster without killing one.
Two of them may take some time. Tzitzi-Ya-Ku seems need double or triple pts than other monsters to fully research. Another one is Kushala Daora, it barely moves so its footprints and markers are pretty rare. I'd suggest use slingshot to lure him move around before kill him.
Edit: Talk to Chief Ecologist to check your progress. Some monsters have 6 levels (6 stars) and some have 4 levels (4 stars) to full fill. You need to fully get them.
After you update all monster research level from/to Chief Ecologist, the achievement pops.