The mother-of-two was found dead after police were called to an address near Wrotham, Kent following a report of concern for the welfare of a woman.
An investigation into her death, which is being treated as "non-suspicious", "unexplained" and "sudden", is continuing.
Speaking to Phillip Schofield and Holly Willoughby via phone, Levine described Geldof as a "very entertaining" and "headstrong" woman.
He explained how she was determined to be recognised for more than simply being the daughter of Irish singer Bob Geldof and the late Paula Yates.
"She very much had her mind made up on what she wanted to do and what she wanted to say and how she wanted to be perceived by the public," Levine said.
"She had difficulties with trying to get that message across for a while - if you're born to a father like Bob Geldof, there is an inevitability that your life is always going to be played out in the spotlight and after the death of her mother that certainly intensified."
Journalist Louise Gannon, who knew Peaches from a young age, recalled how becoming a mother to her two sons, Astala, who will turn two later this month, and 11-month-old Phaedra, changed her as a person.
She was a real mix of her mum and dad...she was just completely bursting with life.
Louise Gannon, journalist
"She could be a real little monkey in her wild days," Gannon explained. "But she always was this girl with absolute attitude and I saw her go from being this rebel, this very troubled girl, and then she became a rebel who found a cause when she became a mum.
"She was a real character, everything she did, she did 100% and she did it with passion."
Mr Levine added: "(She was) somebody who overcame adversity to go on to do great things."
Following the news of his daughter's death, Bob Geldof said he and his family were "beyond pain".
In a family statement on behalf of sisters Fifi Trixibelle and Pixie and younger half-sister Tiger, Sir Bob said: "Peaches has died. We are beyond pain. She was the wildest, funniest, cleverest, wittiest and the most bonkers of all of us. Writing 'was' destroys me afresh. What a beautiful child."
"We loved her and will cherish her forever. How sad that sentence is.
"How is this possible that we will not see her again? How is that bearable?"
I extend my deepest sympathies to Bob Geldof and his family on the sudden and untimely death of his daughter Peaches.
Michael D. Higgins, Irish President
On Tuesday, Geldof's older sister Fifi posted a picture on Instagram of the pair as children, along with the message: "My beautiful baby sister...Gone but never forgotten. I love you, Peaches x."
Meanwhile tributes to Geldof began to pour in after news of her death emerged.
Friend Lily Allen said: "My thoughts are with Peaches' family at this awful time. I hope they get to grieve in peace. Peaches, rest in peace gorgeous girl."
Irish President Michael D. Higgins, who is currently on an official state visit to Britain with his wife Sabina, has also offered his sympathies to Bob Geldof and his family.
He said: "This is such a difficult cross to bear for any family and all of our thoughts are with Peaches' family and friends at this time.
"Sabina and I were due to meet Bob Geldof while on the state visit and we are thinking of him at this time of immense loss."
A post mortem on Peaches Geldof will be carried out on Wednesday. A decision on whether to hold an inquest will be made depending on the final results, which could take several weeks.