Here is the full list of all 54 Modern Combat 5 achievements.
As a Heavy, perform a killstreak of 10 in multiplayer matches
Complete multiplayer matches: 250
Complete campaign missions: 28
Kill a VIP with only one shot
Revive teammates in the campaign: 5 time(s)
Activate Berserk in the first 10 seconds of a battle in multiplayer
Kill enemies using grenades: 50
Join a squad
Win multiplayer matches: 50
Complete multiplayer matches: 50
Kill enemies in the campaign: 500
Capture flags in multiplayer matches: 1
Complete campaign missions: 13
Perform headshots: 500
Perform a kill with a shotgun while playing as a Heavy in multiplayer
Kill flag carriers in multiplayer matches: 1
Kill a flag carrier with just one shot
Win multiplayer matches: 10
Upgrade all the skills for one class
Kill enemies in the campaign: 250
As a Sniper, perform a streak of 5 headshots in multiplayer matches
Kill the enemy shooting at the Decoy in multiplayer
Perform headshots: 100
Complete campaign missions: 5
Perform headshots: 10
Create a squad
Kill enemies in the campaign: 50
Break all the vases in Venice
Destroy a vehicle with a revolver
Kill a flag carrier with a grenade
Win multiplayer matches: 250
Complete the tutorial
As an Assault, perform a killstreak of 5 using grenades in multiplayer matches
Kill a VIP 10 seconds before the multiplayer match is over
Kill a flag carrier 10 seconds before the multiplayer match is over
Perform a kill with an assault rifle while playing as a Recon in multiplayer
Kill enemies using grenades: 250
Kill enemies using grenades: 500
Revive teammates in the campaign: 25 time(s)
Revive teammates in the campaign: 50 time(s)
Kill a VIP with a knife
Kill flag carriers in multiplayer matches: 50
Kill flag carriers in multiplayer matches: 100
Win multiplayer VIP matches: 10
Win multiplayer VIP matches: 50
Win multiplayer VIP matches: 250
As a Recon, perform a killstreak of 5 using only melee attacks in multiplayer matches
Kill VIPs in multiplayer matches: 10
Kill VIPs in multiplayer matches: 50
Kill VIPs in multiplayer matches: 250
Completely upgrade the skills in all classes
Capture flags in multiplayer matches: 50
Capture flags in multiplayer matches: 100
Complete multiplayer matches: 500