Here is the full list of all 21 Combat Core achievements.
Complete Arcade with Bruizer (Any Difficulty)
Complete Arcade with Pugi (Any Difficulty)
Complete Arcade with Skatie (Any Difficulty)
Complete Arcade with Jondo (Any Difficulty)
Complete Arcade with Ember (Any Difficulty)
Complete Arcade with Mist (Any Difficulty)
Complete Arcade with Kayo (Any Difficulty)
Complete Arcade with Squaz (Any Difficulty)
Complete Arcade with Kizuna (Any Difficulty)
Complete Arcade with Zetto (Any Difficulty)
Complete Arcade with Ares (Any Difficulty)
Complete Arcade (Pro)
Complete Arcade (Champion)
Complete Arcade Under 8 Minutes (Rookie Difficulty)
Complete Arcade Under 12 Minutes (Pro Difficulty)
Complete Arcade Under 16 Minutes (Champion Difficulty)
Complete Arcade with a Custom Character (Any Difficulty)
Create a Custom Character
Complete an online match
Complete Arcade (Rookie)
Save a Custom Core