Here is the full list of all 10 Crash The Core achievements.
Welcome New Recruit!
Defeat Your First Boss!
Take Down the World Boss Sol.
Take Down the World Boss The Reaper.
Take Down the World Boss Brax.
Get Prestige Level 2.
Get Prestige Level 3.
Get Prestige Level 4.
Get Prestige Level 5.
Get Prestige Level 6.