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.
Defeat a boss using the METAL SLUG ATTACK
Clear the game using no continue
Clear FINAL MISSION
Clear MISSION 5
Kill 1,000 enemy soldiers
Free 100 POWs
Throw 200 grenades
Free 50 POWs
Play in MULTIPLAYER
Clear MISSION 4
Throw 50 grenades
Clear MISSION 3
Kill 200 enemy soldiers
Clear MISSION 2
Clear MISSION 1