Acquire the Drive ability, then during any objective based fights, complete the fight with a Drive activated and Sora will retain the transformation appearance for the story event that follows.
This battle is a bit difficult because Axel has long range attacks. Since he launches most of his attacks straight ahead, approach him from the side. When his body envelopes in flames, this signals that a strong attack is coming, so get some distance between you and him, and get ready to avoid it.
At the start of the battle, you will find that your weapon has no effect on Dusk. However, after a short while an intermission sequence will play and you will gain the Keyblade, which will allow you to damage him. It should only take 5 hits to defeat him.
At the start of the battle, a dark energy will surround Roxas. Then, the screen will fade to white, and Roxas will be trapped in the air. When Flailer approaches and is about to attack, use the Reaction Command to break free and attack him. Then, Flailer will rush up and grab you. Use the Reaction Command when it flashes to knock him off the platform. The Station will turn upside down and Roxas will slide off, using his Keyblade to hang on. Flailer will generate a sphere of dark energy and launch it at you. As it approaches, use the Reaction Command and Roxas will throw his Keyblade through the attack and knock Flailer down. Next, target his head and start attacking him before he stands back up. When he stands up, he will begin to do various attacks, which includes summoning smaller enemies. Use the Reaction Commands to reach his head, and try to avoid his attacks. Eventually, his attack pattern will reset and you will go through the Reaction Command sequence again. If you take a lot of damage, make sure to use your potions.
Seifer will walk forward and slash at you. Run up to him and begin attacking him before he starts attacking. He will also leap forward and do a spinning slash which is more difficult to avoid; try blocking his attacks with Reflect Guard by pressing Square. He will go down after about 13 hits.
Early on in Kingdom Hearts 2, Sora receives new clothing, replacing his rather silly looking attire from the original game. At the same time, he earns the ability to use Drives which, among many other things, further alter his appearance while activated. Interestingly enough, there is a simply way to extend the length of a Drive's visual transformation, so as to reflect in cut-scenes as well: During any of the game's numerous objective based fights, simply complete the event with a Drive activated and Sora will retain his modified appearance for the story event that follows.