AOF2 was never intended it to be a KOF tournament organized by Geese.
Besides, the AOF2 story was changed in KOF 96 so that Ryo would not be much older than Terry (Refer to any AOF2 ending and the overall Fatal Fury storyline).
Anyways, here's the story.
Yuri reveals to Ryo and Robert that Mr. Karate is really Takuma. Takuma explains that Yuri was kidnapped by Mr. Big when Takuma refused to cooperate with Mr. Big's cartel, and then Mr. Big had used Yuri as bait to get Takuma to cooperate. Fortunately, Ryo and Robert managed to save both Yuri and Takuma, and they begin plotting the ultimate demise of Mr. Big and his cartel.
Despite Ryo's objections, Takuma decides to teach his daughter Kyokugen karate.
King had a steady relationship with Ryo, and Robert paid for Jan's (King's brother's) medical fees. (Although this story would change in 97 as Chizuru and not Robert paid for Jan's medical fees)
Eiji just hates all karate in general, but grew a bigger hatred for Kyokugen after the events of AOF2 (this was taken from KOF 95).
Crawley, retired after AOF1, decides not to come out of retirement despite the fact that his country needed him.
Lee retires, apparently after discovering a new medicine.
The rest of the stories are somewhat obscure to me. For example, Temjin's story has nothing to do with any other character.
In the end, the Kyokugen family (and Robert) discover that they had just done the major gruntwork for a certain crime boss named Geese Howard. Mr. Big was a threat to Geese's power (which also explains their 96 team ending) and wanted him eliminated. Seeing how the Kyokugen school defeated Mr. Big and took him out, Geese took the opportunity to try and let the Kyokugen school join him. But for the obvious reasons (Ryo had claimed that he had known of Geese and how his fater hated him), the Kyokugen school disagreed. Geese proceeds to fight them only to be harshly defeated. The Kyokugen family wanted to finish off Geese, but Geese's underlings created a distraction, allowing Geese to escape.
In the original storyline, these activities would attract the attention of one Jeff Bogard, who, of course, Geese would later kill. In the 96-changed storyline, Geese just simply escapes, period.