This requires you to kill 250 enemies in ANY of the three games and across them. In other words, you'll definitely get the achievement simply by playing, but if you are a completionist, it will pop-up in the first game. Shepard has to be the one taking the final shot.
UPDATE 17.05.2021: Please do not downvote the solution because the achievement is buggy! :(
Today's update from BioWare should help you understand that the fix to the achievement was needed:
https://www.ea.com/games/mass-effect/mass-effect-legendary-e...