There is no in-game indicator for this in regards to which characters you have completed tournament mode, so I recommend going systematically from Master Shifu or writing a list you can cross off
DLC characters count towards this total, you may continue (should you have continue amulets) and you can do so on the easiest difficulty
See the solution for Master Panda/Master Mantis for how to gain "online" XP as you work through these and work on those achievements at the same time