In most KoFs on the Neo, the ending sequences/stories depend on your team lineup, not on the way you beat the game (with/without continues/continue service, etc.). You normally can't unlock secret charas by beating the game, you'll need some sort of code/secret for that to work, like holding start while selecting Chris in KoF98 to play as Orochi Chris, etc.
However, in KoF99 and 2k the scoring system works different. Instead of a score you have a Battle Ability that increases or decreases depending on fighting style, ending a battle with a (S)DM, only using the first chara of your team (Straight win), etc. If your BA rating raises beyond a certain level after successfully completing the 5th match, you will face Kula Diamond ("INVASION"), and beating her has an effect on the endings, the dialogues will be slightly different. Using continues/continue service in these games decreases your BA so that it may be not high enough to get to fight Kula. In a way, continuing DOES affect the endings in this case but you'll still see them even with 0 BA.
In KoF99, having a high BA rating before the 6th round and using a non-edited team will give you an additional fight against Real Kyo or Iori after you have beaten Krizalid but AFAIK it has no effects on the endings. I don't know wether you can use continues to get to them or not.
Ah, and there are charas/features to unlock in some KoF ports on DC (like the Puzzle Fighter mode in KoF2k1), but even for that you just have to beat the game one way or another, you don't have to beat it on one credit IIRC.