Dinos level up 59 times, regardless of what their starting level is.
Being near a dino, petting a nearby Lystrosaurus, riding a dinosaur while someone is crafting (or you), killing things with your dino, and harvesting with your dino gives it XP.
Or just go to single player, ride a dino, use the command "addexperience 999999 1 1" and the achievement will pop when you put in the stats for your dinosaur.