Here is the full list of all 45 Beat Champion achievements.
Hit 95% of notes (or better) on Easy
Get at least 75/100 strength ranking on any song
Complete a Tutorial
Hit 95% of notes (or better) on Medium
Hit 95% of notes (or better) on Hard
Hit 95% of notes (or better) on Champion
Get a perfect strength ranking (100/100) on any song
Beat every song on every difficulty (95% or better)
Earn a score of 0 on any song