(Be sure to check out my good friend Steve's on Fatalities guide and Brutalities guide if you're looking for an easy to use resource on busting some heads.) Cassie Cage
There seems to be some RNG involved in getting some costumes. Some costumes have methods we've yet to whittle down completely yet, but the most common theories and tried methods are listed. Not all costumes can be obtained by actually playing Mortal Kombat X - a few require you use the Mortal Kombat X Mobile app to either link accounts or actually play the mobile game. Every character has one costume they can get just by clearing the Klassic Tower, and some have others they can get by completing other tasks in-game.