This story has been updated.

NEW YORK – Sketching a vision of a more hopeful, inclusive America that takes care of its own while taking on big challenges such as climate change, Hillary Rodham Clinton said Saturday she is running for president to be the champion the country needs now, as well as its history-making first female commander in chief.

Clinton launched her campaign in earnest with a 45-minute address that melded her personal biography as an advocate for women and children with promises of policy changes she promised would make the country fairer for all. It was the first time since entering the race in April that Clinton made a sustained case for her second run for the White House and told voters why she thinks she deserves to win.

"I may not be the youngest candidate in this race, but I'll be the youngest woman president in the history of the United States,” she told a crowd of cheering supporters on the southern tip of Roosevelt Island, with a stunning East River view of the Manhattan skyline in the background, the United Nations building sparkling in bright sunshine behind the podium.

Clinton began her move from small roundtable-style events to more traditional rallies with a speech that was heavy on biography and lighter on policy. The markedly personal address focused on why Clinton wants to run and what would define her presidency if she wins.

From roundtables to rallies, how will Hillary Clinton's campaign fare as it ramps up into its next phase? (Julie Percha/The Washington Post)

Framed around the story of how Clinton’s late mother, Dorothy Rodham, emerged from a childhood of mistreatment without losing her faith in humanity, the speech laid out how Clinton drew lessons about hope, perseverance and kindness from her mother’s example.

"My mother taught me that everybody needs a chance and a champion. She knew what it was like not to have either one," Clinton said.

"I wish my mother could have been with us longer. I wish she could have seen Chelsea become a mother herself. I wish she could have met Charlotte. I wish she could have seen the America we’re going to build together," she added.

Clinton has already written and spoken at some length about her mother’s generosity and resilience, and the example she set for a daughter who had far greater economic and educational prospects. Saturday’s speech was intended to expand on the notion that every child needs a champion.

The emphasis on Clinton’s personal story is part of an attempted rebranding of Clinton’s abiding image as an efficient, and sometimes chilly, policy wonk. She talks with glowing grandmotherly pride about 8-month-old granddaughter Charlotte at nearly every campaign appearance, and frequently invokes her mother’s example of perseverance.

Hillary Clinton delivered her official launch speech at a campaign rally in New York on Saturday. (AP)

A gauzy biographical video released by the campaign Friday cast Clinton as a lifelong warrior for children and families, with photos of Clinton from throughout a 40-year career in advocacy and public service.

"Everyone deserves a chance to live up to his or her God-given potential," Clinton says in the video. "That's the dream we share. That's the fight we must wage."

The speech setting in a New York City park honoring former president Franklin Delano Roosevelt was a nod both to Clinton’s adopted home state, which she represented in the Senate for eight years, and the legacy of Roosevelt’s ethos that America should be free from want and free from fear.

"It is be here in this beautiful park dedicated to Franklin Roosevelt’s enduring vision of America, the nation we want to be," she said.

Clinton invoked Roosevelt’s social and economic programs as she described what she calls the erosion of middle-class expectations and protections in the last generation. She will say that American success should be measured by how well ordinary families are doing, not by the wealth of a few, her campaign said.

The speech had a frequent refrain: “It’s your time.”

"It's America's basic bargain: if you do your part, you ought to be able to get ahead," Clinton said. "Democracy can't be just for billionaires and corporations. Prosperity and democracy are part of your basic bargain too," she added later.

"I'm not running for some Americans, but for all Americans," she said. "...You brought our country back. Now it's time -- your time -- to secure our gains and move ahead."

The economic agenda Clinton has begun to lay out is aligned with policies of the progressive Roosevelt Institute in New York. Institute staff have met several times in recent weeks with Clinton campaign staffers as the campaign readies a series of policy speeches Clinton will give in coming weeks.

Much of that discussion has focused on financial reform and corporate governance issues, said Roosevelt Institute president Felicia Wong.

Clinton’s perceived coziness with Wall Street titans has caused some ambivalence among progressives and it is not clear how far she intends to go to tighten regulations on banks and other financial institutions.

The institute -- which was involved in the creation of the Four Freedoms Park where Clinton speaks Saturday, and it shares board members with the nonprofit conservancy that runs the park -- recently published a report pinning blame for middle-class economic decline on what it calls failed models, such as supply-side or trickle-down economics.

In her speech, Clinton condemned what she called Republicans’ “top-down economic policies that failed us before."

"These Republicans trip over themselves promising lower taxes for the wealthy and fewer rules for the biggest corporations without regard for how that will make income inequality even worse," she said. "We’ve heard this tune before. And we know how it turns out."

But there wasn't a State of the Union-style list of priorities and policy specifics. Clinton will give speeches and hold a series of events through the summer and early fall that address specific economic and other policy matters, aides said.

“The speech is about her. What I think is the diagnosis of the problems in the country; this is my vision of where I want to take the country; here are my solutions,” Palmieri said.