Ms Warren, 69, a senator from MA since 2013, became one of Mr Trump's fiercest critics during the 2016 presidential race and they have continued to exchange biting insults during his presidency.
The exploratory committee is an organization established to help determine whether a potential candidate should run for an elected office. He pushed her into the error of taking that DNA test.
Said Warren in the clip: "I've spent my career getting to the bottom of why America's promise works for some families, but others, who work just as hard, slip through the cracks into disaster".
Trump's use of the name Pocahontas refers to a 17th century Native American woman associated with the British colony in Jamestown, Virginia, and was aimed at drawing attention to a controversy over her heritage raised during Warren's 2012 Senate race.
Republican National Committee Chair Ronna Romney McDaniel wasted no time putting Democratic Sen. "Americans will see her for what she is: another extreme far-left obstructionist and total fraud".
Warren was re-elected for a second term as Massachusetts' senior senator in November, easily defeating Republican challenger Geoff Diehl.
The paper also reports that the former Harvard bankruptcy-law professor plans to start building momentum in early voting states, such as Iowa, in the coming weeks. What does her home state polling tell us about a possible 2020 matchup?
"If we organize together, if we fight together, if we persist together, we can win", Warren ends the video, making a subtle reference to a moment on the Senate floor in 2017 that went viral. This represents "a marked departure from 2016, when left-leaning grass-roots groups were hungry for Warren to take on the more establishment Hillary Clinton in the primary as their candidate".
The video, which runs for almost four-and-a-half minutes, touches on Warren's family's military tradition, her parents' struggle to provide their children with opportunities and her career fighting for the marginalized in a political system that has been rigged to favor the wealthy.
Elizabeth Warren (D-MA) announcing she is forming an exploratory committee for a presidential run, CNN's Harry Enten said her performance during the 2018 midterms is a good indicator to how she could perform in 2020.
In her first campaign blunder before even announcing her exploratory committee, Warren's rollout of the DNA test results was met with mockery and backlash. "There's no rush to do it". You know it. I know it. In one such episode, she turned an insult from Senate Majority Leader Mitch McConnell (R-Ky.) - "nevertheless, she persisted" - into a liberal rallying cry. Other Democrats who are supposed to jump on the campaign trail before the mid-point of 2019 include Kamala Harris, Cory Booker, Kirsten Gillibrand, and former New York Mayor Michael Bloomberg. But while Warren may in theory be the flawless candidate to heal that rift, she's still a woman - and that remains more of a liability than we may want to believe.