Here is the full list of all 15 METAL SLUG achievements. It takes around 6-8 hours to unlock all of the achievements on Windows.
Clear the game using no continue
Kill 1,000 enemy soldiers
Free 100 POWs
Defeat a boss using the METAL SLUG ATTACK
Clear FINAL MISSION
Free 50 POWs
Clear MISSION 5
Clear MISSION 4
Throw 200 grenades
Clear MISSION 3
Kill 200 enemy soldiers
Clear MISSION 2
Throw 50 grenades
Clear MISSION 1
Play in MULTIPLAYER