Cosmo "Oz" Potter lived a hard life from a young age. She had the memory of what she knew was her mother and a pale snake-like man entering and some green lights. She knew the green light was not friendly when her mother went down in a heap, after getting hit with the same light, she could testify that it was painful and she would rather never have to repeat that again.
After that night she was place in the place that would become her metaphorical and literal prison and hellhole. Whoever that old white bearded bastard was on her list of people she was going hurt. Her relatives-she refused to call them family- were disgusting people that she would have rather not meet in another world. Really it was almost as if they were trying to somehow imitate animals. Of course she never told them to their face, but she did silently in the comfort of her cupboard. She was sure that the way she was treated was wrong and she was questioning the sanity of all the adults near her when they turned a blind eye to the obvious abuse and neglect. Whatever she cursed them too.
Having a quick mind like she did (something the Dursleys disapproved of since they found out in nursery care), it was only logical for her to try and find ways to make her life just a bit easier. Reading became essential, as did being able to blend in and be overlooked, and pickpocketing. She felt bad about the last part at first, but she figured that if others refused to help and care for her then why should she care about them?
Oz also didn't trust people easily, -you could thank her relatives for that-, once a girl tried to befriend her on the basis of a dare. Her being able to blend in helped her overhear the dare as it was being made. She was approached the next day by the girl, Oz learned that day that people weren't they tried to make themselves out to be. People tried to hide their true intentions, thus shouldn't be trusted, at least the people in Surrey and others would have a little leeway to her mistrusting nature.
By her fifth year alive (really it was just trying to stay alive with the neglect and minor beatings she was treated to), she was under no illusion that her life would get better in the 'care' of her relatives. It would only get worse. Thus she made sure that when she was not allowed in the house (not home, never home), she had little stashes of food and water and other necessities in different places in Little Whining and just a few places further than that.
Oz was always aware of the animosity between her aunt, uncle, and cousin. Who wouldn't know when the other would do everything to remind them that they hated them and were different in the sense that they were normal and she was not (and for that she was grateful, because being ANYTHING like...them.. was just disgusting).
However knowing how much her relatives despised her and seeing the length they would go to get rid of her were too different things. She discovered that they would go great lengths when they dragged and dumped her in Italy, three months after she turned five.
She swore that she would survive to one day hurt the Dursleys and the bearded bastard she still remembered. If anything this served as motivation to survive and become better. (Fate cackled at the turn of events and how this would destroy the plans a certain old man had. She would help little Oz remain hidden from that man to avoid having him find her. No way was she letting little Oz complete her destiny the way that old fool wanted. Destiny only watched and smirked at the thought of a completely 'Girl-Who-Lived' showing up, especially to one that was about to survive and grow in Mafia Italy. Her smirked grew larger, the British Magical World wouldn't know what hit them.)
(The British Magical World felt a sense of foreboding of the future that same day)