Get a perfect on all bosses when you fight them. I don't know if it has to be when you run their stages or during the boss rush where you have to fight them all again or if both counts incase you failed the first time.
Credits goes to sonicboom510 on twitch.tv for making it that far.
That was a tool assisted run up until the Akuma encounter btw. I was talking to another person who got to Akuma in the legit way. He said he only had 5 perfects (none at the boss rush) and was able to reach Akuma.