Each Adventure level has three badges you can earn: Level Complete, Spirit Score, and Spirit Time. You'll automatically get Level Complete, but the other two are mostly skill-based (with a side of luck). If you fail to get one or more badges, you can press B after the level to return to the Menu. From there, select the level and try again. You can also get better results by coming back to levels after you've unlocked more Spirit Animals and leveled them up with badges.
The soonest you can get this Achievement is after completing 1-10, but you can also opt not to replay any levels you missed badges on and just get all three badges in a few later levels.