It is the latest salvo in a bitter war of words between Mr Cummings, the PM's former top aide, and Matt Hancock over the handling of the pandemic.
Downing Street did not deny the authenticity of the messages.
But the PM's official spokesman insisted Mr Johnson has full confidence in the health secretary.
In a 7,000-word blog, Mr Cummings accused Mr Hancock of trying to rewrite history during a select committee hearing last week which was examining how the government has managed various aspects of the crisis.
He also took aim at Mr Johnson, claiming he had "lied about failures" during the government's initial response to the pandemic.
"If No 10 is prepared to lie so deeply and widely about such vital issues of life and death last year, it cannot be trusted now either on Covid or any other crucial issue of war and peace," Mr Cummings wrote.
Mr Cummings told MPs three weeks ago that the health secretary had lied to the prime minister about testing hospital patients before they were released into care homes at the start of the pandemic - something firmly denied by Mr Hancock.
A source close to Mr Hancock said: "The health secretary answered many of Mr Cummings' claims at a lengthy session in front of MPs.
"Mr Cummings has still failed to provide evidence to substantiate his claims.
"The health secretary continues to work with the prime minister on the vaccine programme and getting us out of this pandemic as quickly as possible."
Mr Cummings, who was forced out of Downing Street last year, claims the version of events given to the committee by Mr Hancock was "fiction".
The former aide appears to have kept WhatsApp messages from his time in government and selected six to share with readers of his blog.
One of them appears to show an exchange between Mr Johnson and Mr Cummings on 27 March last year.
Mr Cummings highlights the ramping up of testing capacity in the US and criticises Mr Hancock for saying he was "sceptical" about meeting a target.
Mr Johnson purportedly responds: "Totally [expletive] hopeless."
Mr Cummings also published another private message about the struggle to procure ventilators for Covid-19 patients.
"It's Hancock. He has been hopeless," a contact appearing to be Mr Johnson replies on 27 March last year.
In another message, on 27 April last year, the prime minister appears to call the situation around personal protective equipment (PPE) "a disaster" and alludes to diverting some responsibilities to Cabinet Office minister Michael Gove.
"I can't think of anything except taking Hancock off and putting Gove on," Mr Johnson apparently adds.
Mr Johnson's official spokesman said he was "not going to be drawn on these claims", adding that the prime minister has "worked very closely with the health secretary and continues to do so".
Labour's shadow health secretary Jonathan Ashworth told Mr Hancock in a Commons debate that he had been "forever branded Hopeless Hancock by his own leader".
The party's deputy leader Angela Rayner said a public inquiry into the pandemic promised for next spring was now needed "as quickly as possible".
Mr Cummings suggests in his blog that any inquiry would not "start for years" as it was "designed to punt the tricky parts until after this PM has gone".
"Unlike other PMs, this one has a clear plan to leave at the latest a couple of years after the next election, he wants to make money and have fun not 'go on and on'," writes Mr Cummings.
But with the Conservatives riding high in the opinion polls, there is no incentive for the party's MPs to move against Mr Johnson, despite the "systemic incompetence" surrounding him, he adds.
Downing Street rejected the claim that Mr Johnson planned to stand down after the next election, saying: "The PM has actually been asked this before and said it's utter nonsense, so that still stands."
Let's start with a quick political quiz.
Q: Who has the safest job in cabinet?
A: Matt Hancock
A: Because Dominic Cummings wants him sacked.
Boris Johnson's spokesman said the PM had full confidence in the health secretary.
He could say little else - as he won't be sacking him at the urging of a disgruntled former aide.
If he did, that would suggest that Dominic Cummings was right, about what he thinks were failures over PPE, ventilators and testing, early in the pandemic.
And if Mr Hancock goes now, Labour would ask why he wasn't dismissed sooner.
Mr Cummings makes serious allegations about the handling of the Covid crisis.
But by continually suggesting that Matt Hancock is the very personification of government errors, he may be making it a little too easy for his detractors to portray his interventions as a grudge match.
And by denouncing not just the Department of Health and Social Care but the Cabinet Office too, you get the impression that Mr Cummings felt that very few people could come up to his standards.
Mr Cummings's own credibility, of course, still remains dented due to last year's lockdown drive to Durham.
But his latest dossier does raise questions of ministers well ahead of the formal inquiry, and provides ammunition to an opposition which has questioned the pace at which the government moved.
Giving evidence to MPs last week, Mr Hancock said he had seen no evidence to suggest any medical staff had died because of a lack of PPE.
In his blog, Mr Cummings accuses the health secretary of trying to blame NHS England chief executive Sir Simon Stevens, Chancellor Rishi Sunak and the Cabinet Office for last April's "PPE disaster".
"The lack of PPE killed NHS and care home staff in March-May," he writes.
He claims Number 10 and Mr Hancock had "repeatedly lied about the failures last year" and accuses them of now trying to "rewrite history", with the prime minister "encouraging ministers to give false accounts to Parliament".
Mr Cummings goes on to criticise Mr Johnson's management style.
In one section he contrasts Mr Johnson's approach with that of Foreign Secretary Dominic Raab, who stood in for the PM when he was in hospital with coronavirus.
He said meetings with Mr Raab at the helm were "less pleasant for everybody but much more productive". He said this was because Mr Raab "can chair meetings properly instead of telling rambling stories and jokes".
He went on to say that "as soon as things get 'a bit embarrassing' [Boris Johnson] does the whole 'let's take it offline' shtick before shouting 'forward to victory', doing a thumbs-up and pegging it out of the room before anybody can disagree".
Downing Street said it was "not going to engage" with this claim.
Mr Cummings published the alleged leaks on Substack, an online platform that allows people to charge for newsletters.
He has said he plans to charge subscribers for insider information on subjects other than the pandemic.