Princesses of Heart must have a pure heart full of innocence and purity. They can't be Princesses of Heart unless they have commited nothing harmful of anyway.
A theory says that Kairi was a princess in a far land, or soon will be, until heartless attacked that land. So Kairi escaped and came to Destiny Islands.
Areil wasn't born human. She might not have one heart, but maybe two. Ariel also stole her fathers trident for her own selfish needs, so she was guilty, or not innocent.
Princesses of Heart don't nessecerily have to do good deeds,but musn't do bad deeds.
Did Alice do something right? No, but did she do somehting wrong, no!
I hope this clears up some confusion