What the achievement asks
Reach Renown 20 with the Dream Wardens. Five points, account-wide, both factions.
What renown actually costs
Renown with a Dragonflight faction comes from almost everything you do in its zone: world quests, dailies and weeklies, rares, events, and the faction's own questlines. There is no single fast route and nothing you can buy to skip it.
What there is, is a floor. Renown never resets and it is account-wide, so every character you play feeds the same bar, and a faction you start late pays more per activity than one you have been grinding since launch. Coming to this now is considerably faster than it was at the time.
Do the weekly Aiding the Accord every week without fail. It asks you for 4,000 reputation from whatever you were doing anyway, and it pays out a chest that gives 500 renown reputation to every main faction at once. Nothing else in the expansion moves six bars with one turn-in.
Where its renown comes from
The Dream Wardens are the Emerald Dream faction, and almost all of their renown comes from the zone's two big events rather than from ordinary questing.
The Superbloom is the large one, and running it well matters twice over: its reward satchel quality is Super Duper Bloom in the same meta, and the bonus you build up by doing the route's small activities is what raises both. The Dreamsurge, when it lands in a Dragon Isles zone, is the other, and its weekly quest is Dream Shaper.
The Emerald Dream campaign itself pays renown as you go, and it is Defenders of the Dream in another branch. Do it first: it is the single largest lump of Dream Wardens renown available, and it opens the zone's other content at the same time.
Twenty is the joint-lowest of the six renown requirements, and the Emerald Dream is compact enough that a few Superblooms and the campaign cover most of it.
Where it fits
One of twelve under Oh My God, They Were Clutchmates, a piece of A World Awoken. Renown is the slowest currency in the meta and the one with the least to do with skill, so start every faction early and let the weeklies carry them.