Here is the full list of all 15 Double Dragon Trilogy achievements. It takes around 4-5 hours to unlock all of the achievements on Windows.
Complete DD2 in Arcade and expert mode
Have more than 10 game over
Complete DD2 Story mode
Complete DD3 in Arcade and expert mode
Complete DD3 Story mode
Start all 3 games
Use all the weapons of the game (DD1)
Complete DD1 in Arcade and expert mode
Start a Multiplayer game (local or online)
Make 100 hurricane kicks (DD2)
Beat the Terminator Boss in 2nd stage (DD2)
Use the travelator in stage 2 (DD2)
Make a 50 000 pts score (DD1)
Complete DD1 Story mode
Complete a lvl without taking a hit (DD1)