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