Protesters, some holding fliers or wearing T-shirts with images of missing people, climbed the front steps and pushed their way into the building.
Some attempted to force their way past security personnel to the upper floors at the headquarters of the Royal Borough of Kensington and Chelsea.
One man approached a TV camera, saying, "We want answers and justice." People did not favor violence but were desperate for information, he said.
Police said Friday that 30 people were confirmed dead in the fire that broke out early Wednesday, but residents have said they believe the number could be much higher.
A second protest began later at Britain's Home Office, which oversees fire prevention and policing nationally. Organizers handed out fliers that read, "Your anger must be heard." The protesters eventually made their way to busy Oxford Circus, where they staged a sit-in.
Theresa May, Britain's embattled Prime Minister, visiting nearby, was bundled away by security guards after protesters converged on the street. May has already faced criticism for failing to meet residents when she made a low-key visit to the scene Thursday.
The UK government, which has already promised a full public inquiry, announced a fund of 5 million British pounds ($6.4 million) to help those affected by the blaze.
London Mayor Sadiq Khan demanded that the government publish a list by the end of Friday of other tower blocks checked by investigators. In a letter to May, he listed a litany of residents' concerns, including a lack of information about missing relatives, the chaotic response of the local council and worries over safety in other tower blocks.
"They feel the government and local council haven't done enough to help them in the aftermath of this horrific incident, or to provide answers to their increasingly urgent questions," Khan wrote in the letter.
Residents are furious that many of the victims have not yet been found or identified, and they believe authorities have been downplaying the number of victims. They say their long-standing concerns over fire safety at Grenfell Tower were not adequately addressed.
Anger on the streets
One of the speakers at Friday afternoon's protest in West London, Mustafa Al Mansur, told reporters that people were unsatisfied with the official response to the fire.
Al Mansur said there are still many unknown details, such as information about the contractors who worked on the building's renovation in 2015 and 2016.
He also said the tragedy underscores some of the lingering issues of inequality that have roiled Britain in recent years.
"This is the richest borough in the UK but we have a building where some of the poorest people live," he said in an interview. "The safety measures are totally inadequate."
He said the council spent money on the recent renovation merely "to decorate the exterior" so that it would "look better for the rich neighbors."
He said there needs to be an independent inquiry, adding, "People are calling it a corporate manslaughter."
At the Home Office, protesters stood outside with signs and made speeches. The crowd included a heavy presence of union members, representatives of Socialist and Marxist groups, and students.
The crowd then began marching toward 10 Downing Street, the official home of the British Prime Minister. They gathered at the gates that bar access to the street and kept up a barrage of chants: "Blood on your hands" and "May must go."
When the protesters moved on to Oxford Circus, they staged a sit-down protest amid commuters and shoppers. The protest brought the busy retail area to a standstill. At least one scuffle broke out on heavily-trafficked Regent Street nearby.
Police identify fire's origin
Police have opened a criminal investigation into the circumstances of the blaze. They said they have examined the apartment where the fire started and "there is nothing to suggest the fire was started deliberately
," but a team of senior detectives is investigating whether other offenses were committed.
One London member of Parliament has called for corporate manslaughter charges after learning flammable material was used to clad the building during the recent renovation. No sprinkler system was installed.
Queen Elizabeth II and Prince William, the Duke of Cambridge, visited the area to meet residents and community representatives Friday.
Metropolitan Police Commander Stuart Cundy said there is "a risk that sadly we may not be able to identify everybody." He said 24 people are still hospitalized, with 12 of them in critical care.
Details of victims emerge
The names of those who died are beginning to emerge.
One was Gloria Trevisan, a 26-year-old Italian architect
who was living in London because her family was having financial difficulties at home in Italy.
A lawyer for the family told CNN that Trevisan was able to speak with her parents before she died, telling them: "I am going to heaven. I will help you from up there."
Mohammad Al-Hajali, 23, a refugee who fled Syria for the UK in 2014, was identified Thursday as another victim.
Mohammad and his brother Omar, both students, lived together on the 14th floor.
Omar, 25, survived, but the brothers were separated as fire fighters tried to rescue them from the burning building early Wednesday.
The UK government has promised that all those left homeless by the disaster will be rehoused in the local area. But the government has been criticized for failing to act on recommendations from previous tower block fires.
The Prime Minister is facing particular scrutiny over the role of her new chief of staff, Gavin Barwell, who served as housing minister until he lost his seat in last week's snap general election.
Barwell had told lawmakers that the government intended to review fire safety standards following a fatal fire at Lakanal House, a high-rise building in south London, in 2009. Three women and three children died. In that case, and exterior paneling helped the fire spread.
Jim Fitzpatrick, an MP with the main opposition Labour Party, slammed the Conservative-led government for shuffling its feet on the review.
He said that he and a parliamentary group that he chairs have been pushing for better safety regulations since the 2009 fire.