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