Depends on the game, Amon requirements have gotten progressively easier:
Yakuza 1 requires you to beat all the side stories (making sure you don't fail any of course!). I forget if there is any arena requirements for him in the first game or whether that was just a bonus (I just remember Komaki being a nightmare!)
Yakuza 2 has the most complex set of requirements out there:
All side stories (again, don't fail/miss any which is super easy to do in places), locker keys? (I think), you have to unlock all abilities/moves as you go (some of which are fairly tough to track down), pull off every heat move in the game (including the ones that can only be done in certain story fights and boss fights), defeat everyone in the arena including the rare fighters etc. And you have around 10? hostesses in the game? (I'm sure its between 7-10, its a big number is my point!) AND you have host and hostess club management side stories to complete.
Yakuza 3 is easy, just do the side stories, but Ryu Ga Gotoku 3 is harder since as part of the side stories you need to clear the hostess club management mini game (not in the English version) which can be fairly difficult if you either don't follow a guide or don't know what your doing.
Y4, 5, Ishin, 0 all have you just do all the side stories which is nice and straight forward. Yakuza 6 is hilariously easy as you don't even have to do all the side stories - just do 2 specific ones and then you get to fight Amon (but really fighting him without leveling up heavily is a crazy idea as he's pretty darn powerful still).
I think Kenzan has a more complex set of requirements compared to 3 and up, just less complicated than 2. I still need to go back and fight him in Kenzan, its the one game I've never unlocked his fight, as I got screwed out of one side story.