The Incredible Princess
Vivi's past is first mentioned by Dalton in the Drum Island Arc. She was taken to a meeting between the Counsel of Kings by her father and some of his close personal followers. During the meeting Wapol had been upset by the words of her father and after the meeting he bumped into her on purpose knocking her over. Despite being she stood up and in a mature manner apologized for being so clumsy and knocking into Wapol, impressing Dalton. Afterward, out of Wapol's sight, she cried about how much it had hurt.
The Suna Suna Clan
11 years ago Vivi meet Kohza for the first time. His father had paid the King a visit and Kohza and Vivi met each other and a fight broke out between the two. Kohza's father apologised for his son's action towards the princess of the kingdom that had become their home, however the King recognised it was just a child's fight and took no further action against Kohza or his father.
Later, Kohza invited Vivi in a fight to join his group called the Suna Suna Clan. She fought him again and lost, but gained his followers' respect and was declared the sub leader of the Suna Suna Clan. For many weeks on end she and the others would play together, her father and Igaram were always watching over them to ensure their safety (they followed her). One day, she was kidnapped on route and with the aid of all whose who cared about her, she was rescued. Kohza left to the oasis of Yuba soon afterwards to form a new town.
At some point during Vivi's childhood, she tried to make fireworks out of gunpowder to commemorate the anniversary of Pell joining the guard. Pell slapped her, and told her that if any harm befell her, he would not know what to do. As a way of making up with her, he allowed her to ride on his back as he flew as a falcon, and told her that he trained hard to grow strong so that he could protect the kingdom.
Baroque Works Conspiracy
Then many years later Crocodile of the Shichibukai came to the country. After a period of drought in which the country (all save the capital Alubarna) experienced a lack of water. Everyone claimed the falling rain over the capital was a blessing of the King until the day came some Dance Powder was found being shipped in by the King. Many knew the King would never do this, but the findings were enough to stur many thoughts from the younger citizens.
When the reputation of the King was sullied and a rebellion instigated, Vivi (along with Igaram, the Captain of the Royal Guard) infiltrated the criminal organization "Baroque Works" under the alias "Miss Wednesday" in order to discover the identity of the mastermind and prove the king's innocence. There, she discovered that the villain behind the chaos in her homeland was none other than Sir Crocodile, a member of Shichibukai).
Bringing Hope Home
As she was still ostensibly a Baroque Works Frontier Agent when she first met Luffy, the Straw Hats first treated her as an enemy. After her backstory was revealed (and Nami extorted money out of Igaram), however, they agreed to take her in and bring her safely to Alabasta. Despite being significantly weaker than most of the Straw Hats in combat, Vivi's determination inspires them to continue fighting, and Crocodile is defeated and the rebellion ends.
Final Bid Farewell
During her time on the Going Merry, Vivi grew close to the Straw Hat Pirates, admiring their indomitable spirit in the face of adversity and their kind-hearted nature. She even reflected that she'd like to be a member of their pirate crew, if she wasn't a princess. The feeling was mutual, in fact: Luffy offered to let her join (they even considered her as their nakama), but though Vivi very nearly did, she ultimately (and tearfully) declined the offer, saying that her country needed her.
To avoid Vivi being recognized as their ally, they said goodbye to her, showing the X mark of their crew as a sign of their friendship.
Vivi was last seen enjoying the geothermal baths of Ukkari Hot Spring, owned and operated by Goro (Toto's brother and Koza's uncle). She was later seen when the Straw Hats received their bounty raise. Her father and the others alert her at an article they've in the newspaper about the Enies Lobby attack and how one of her biggest foes is working with the Straw Hats now. Yet despite seeing Miss All-Sunday amongst the crew, she was happy about how they were getting along, saying she trusted Luffy's judgement completely concerning his choice of crew members.
Despite being a princess, Vivi never demands any special attention, nor does she put herself first above others. She often thinks of others before herself, risking her life to help anyone who needs her. She is also extremely brave when she needs to be, going up against powerful enemies to help her friends despite clearly being outmatched. Her selfless attitude and sense of responsibility are what pushes her to carry her country's burden.
When she argues, Vivi has a hard time coming up with insults, usually just saying something of little offense- such as nicknaming Zoro 'Mr. Bushido' and has a habit of biting her bottom lip when she is tense. She often gave little speeches about responsibility while being a Straw Hat, but she does not like public speaking. When she joined the Straw Hat Pirates, her animation changed. Most fans have noted that Vivi looks evil when she was still a member of Baroque Works. A running gag throughout most of the Arabasta Arc is that Vivi constantly forgets to tell the Strawhats about the dangers of the desert, such as giant scorpions and lizards, which Luffy, Zoro, and Sanji quickly dispatch but quickly berate her inability to remember such things beforehand