Hint :
Unlock Burning Batsu:
In story mode select Taiyo Gakuen. Now you have to intentionally lose a round with Batsu to Akira, Yurika, or Zaki. Then Iincyo will replace Batsu after the match. Burning Batsu will appear when fighting Vatsu.

Unlock Powered Akira:
In story mode select Seijyun Syogakuen. Now defeat Wild Daigo without using any team up techniques.

Unlock Roy Bromwell and Tiffany Rose:
To unlock these you have to successfully complete the story mode with the Pacific High School.

Unlock Vatsu:
To unlock Vatsu you have to successfully complete the story mode with Gorin Koukou.

Unlock Wild Daigo:
To unlock Wild Daigo you have to successfully complete the story mode with Gedo Koukou.

Unlock Hyo Imawano and Dark Side Student Council:
To unlock these you have to successfully complete the story mode with all the normal schools.

Unlock Kurow:
To unlock Kurow you have to successfully complete the story mode with the Dark Side Student Council.

Unlock Demon Hyo:
To unlock Demon Hyo you have to successfully complete the story mode with all schools and endings.