In one of his most emotional interviews to date, the Prince told ABC television's Good Morning America program that he thought about his mother every day and hoped she would be "proud" of him.
He also said he hoped he was a "fun uncle" to Prince George and Princess Charlotte and that he "can't wait for the day" he has children of his own.
The Prince, who was just 12 when Princess Diana died, said: "I hope she's looking down with tears in her eyes, being incredibly proud of what we've established, I suppose.
I'm sure she's longing for me to have kids so she can be a grandmother again.
"But I hope, once again, everything that we do privately and officially, that it makes her proud."
He added that losing his mother at such a young age had shaped his life "massively", and said his family would "do everything we can to make sure she is never forgotten".
"I hope that a lot of my mother's talents are shown in a lot of the work that I do," he said.
He also shared his "very happy memories" of visiting Disney World in Florida with his mother.
He said he went on the Space Mountain rollercoaster 14 times and thought it was "absolutely fantastic, the best thing ever".
"So, you know, there's all sorts of places over the world where we were very lucky to have those moments with our mother," he said.
Describing his love for children, he said: "I've got a kid inside of me. I want to keep that. I adore kids. I enjoy everything they bring to the party. They just say what they think."
The Prince said he had the "utmost respect" for the Queen as his grandmother - and his "boss" while he was in the Armed Forces. "She was my boss for 10 years, and I viewed her very much like that.
"Now it's really nice because I can go to her for advice and bend her ear over all the experience she's had over the years."
Prince Harry arrives in Kathmandu in Nepal on Friday, where he is representing the Queen on a visit, and says he wants to thank the Gurkhas for their 200 years of service to the Crown.
The Prince also wants to "shine a spotlight" on the Himalayan country, which was devastated by an earthquake almost a year ago, and encourage tourists to return.
Making his first trip to Nepal, the Prince will spend a night as the house guest of an 86-year-old widow whose husband served in the Royal Gurkha Rifles, at her home 6,000ft up in the foothills of the Himalayas in the Lamjung region.
He will also meet the families of Gurkhas killed in Afghanistan.
Alex Pope, of the Gurkha Welfare Trust, said the Gurkhas have great respect for the Royal family, adding: "It's well known among the Gurkha community that Harry served alongside 1st Bn Royal Gurkha Rifles in Afghanistan in 2008, so Harry is particularly popular."
The Prince saw Gurkhas in action with the Army and has said there is "no safer place" than by the side of a Gurkha.