To unlock Billy and Saiki, you gotta reach a certain score after the 3rd match, and you will encounter either of them (depending on your score).
Billy is fairly easy to unlock. Winning 3 matches with 1 character will certainly grant you the fight.
Saiki on the other hand is a bit tricky. I think you have to reach a score of 200,000 by the end of the 3rd match. Make sure to perform as many target actions as you can (they are the tasks that are written under your health bar), but most importantly, do not lose any round.
The below is story mode related. No major spoilers, but I recommend that you read it after you have gone through this mode at least once.
[Spoilers]:
In the beginning of story mode, you will have to pick multiple scenarios. There is a scenario called "The KOF Teams". When you choose it, you get to pick a team. You will play a match that serves as an introduction to the team you pick. The team you pick, and the outcome of this match don't matter, and you will move forward in the story just as if you have chosen any of other 2 scenarios.
Sadly, the real, full introduction of each team is only available in the game's official site.
[/Spoilers]