Not all of the Emergent Traits have been completed yet, as you can see from the Civilopedia: some have details only for Level 1, meaning that Levels 2 and 3 need to be done. There are also some Traits that go with Emergent Leaders, such as Caretaker, Elder, Strategist, Trader, that don't even list multiple levels, so those are also incomplete.
If I get an Emergent Leader with a Trait that hasn't been completed yet (I always play all random civs), I just start a new game.
For the Traits whose levels have been completed, you basically level up by doing things consistent with that Trait. In my current game, for example, I'm playing as Rizuruk of the Clan of Embers, whose Matriarch Trait gives bonuses to Heroes. From what I can tell, the way I've leveled up is by having my Heroes win battles, basically. With the Amurite leaders with traits like Hydromancer, Pyromancer, etc., you use spells with the corresponding Mana, and so on. Traits like Trader, or Strategist, there's nothing you can do right now, and I'm not sure how something like Trader would level up anyway, maybe by building buildings that increase Trade Routes and Trade Route Yields?
I've actually thought that the Leaders with Elder or Trader should be converted from Emergent Leaders to Minor Leaders, so they can acquire a second Trait, rather than trying to figure out how to level them up. (Strategist, on the other hand, is a Trait that could easily work with additional levels.)
