Here is the full list of all 29 Double Dragon IV achievements.
Clear the 10th floor.
Clear the 20th floor.
Clear the 30th floor.
Complete all Missions.
Complete Mission 6.
Complete Mission 8.
Clear the 40th floor.
Clear the 50th floor.
Clear the 100th floor.
Complete all Missions using Williams.
Complete all Missions using Rowper.
Complete all Missions using Linda.
Complete all Missions using Abobo.
Complete all Missions using Right-Arm.
Complete all Missions using Burnov.
Complete all Missions using Chin-Taimei.
Complete all Missions using Mason.
Complete all Missions using Cody.
Complete all Missions using Jake.
Complete all Missions using either Casey or Shannon.
Complete all Missions using Ashida.
Complete all Missions using Yamazaki.
Complete all Missions using Soeno.
Complete all Missions using Ayumi.
Complete all Missions using Kodani.
Complete all Missions using Chin-Seimei.
Complete all Missions using Ranzou.
Complete all Missions using MysteriousWarrior.
Complete all Missions with many characters.