Here is the full list of all 24 SNKRX achievements.
The base game contains 22 achievements, and there are 2 DLC packs containing 2 achievements.
Win with a full set of Healers
Win with a full set of Mages
Win with a full set of Voiders
Win with a full set of Psykers
Win with a full set of Nukers
Win with a full set of Cursers
Win with a full set of Swarmers
Win with a full set of Conjurers
Win with a full set of Rogues
Win in NG+5 difficulty
Win in NG+1 difficulty
Win with a full set of Warriors
Win with a full set of level 3 units
Win with a full set of Forcers
Win with a full set of Rangers
Win with a full set of Enchanters
Win with a full set of level 2 units
Kill the Cluster
Kill the Forcer
Kill the Swarmer
Kill the Exploder
Kill the Speed Booster
Win with a full set of Sorcerers
Win with a full set of Mercenaries