Here is the full list of all 16 Contra Anniversary Collection achievements.
Complete all games
Complete Super Probotector: Alien Rebels
Complete Probotector
Complete Contra III: The Alien Wars
Clear Route E in Contra: Hard Corps
Clear Route D in Contra: Hard Corps
Clear Route C in Contra: Hard Corps
Clear Route B in Contra: Hard Corps
Clear Route A in Contra: Hard Corps
Complete Operation C
Complete Super C
Complete Contra (JP)
Play each game
Complete Contra (NA)
Complete Super Contra
Complete Contra (Arcade)