In order to lock the character endings you can take one of two courses. The first, which I don't suggest, is completing arcade using each character solo or selecting them first and then using a second character of your choice. This only gives you one ending even if you pick tag, and can be rather frustrating given the end boss.
The other method is that you can fight ghost battles which help you achieve the Tekken Lord rank quicker, all the while choosing the battles in which the rectangle is gold for a gold box which often contains endings. Personally I had a few more extra by the time I finished. If you do this and haven't collected enough endings by the time you achieve Tekken lord, I would suggest playing arcade unless the boss gives you a ton of trouble.
EDIT: As noth1ingness said in the comments below, it appears that you win the ending of the character you face during gold box battles. If you have already beaten them or have their ending it is best to focus on ranking up, though the boxes will still give you gold and / or items if you want them.