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.
Have more than 10 game over
Complete DD3 Story mode
Make a 50 000 pts score (DD1)
Make 100 hurricane kicks (DD2)
Complete DD3 in Arcade and expert mode
Complete DD2 Story mode
Complete DD2 in Arcade and expert mode
Complete DD1 Story mode
Complete DD1 in Arcade and expert mode
Complete a lvl without taking a hit (DD1)
Start a Multiplayer game (local or online)
Use the travelator in stage 2 (DD2)
Use all the weapons of the game (DD1)
Start all 3 games
Beat the Terminator Boss in 2nd stage (DD2)