The former foreign minister is the favourite among the 10 candidates vying to succeed Theresa May, who is stepping down after being forced to delay Britain's departure from the European Union twice.
At a launch event in London packed with senior members of the ruling Conservative party, Johnson insisted that Brexit must happen on October 31, the latest deadline agreed with Brussels.
He said Britain must prepare to leave with no new arrangements if need be, but softened his previous rhetoric, suggesting that this was "a last resort, not something that anybody desires".
However, as a protester outside shouted "Stop Brexit" and "Bollocks to Boris", Johnson offered no details of how to break the deadlock in parliament that prevented May from getting her deal through.
He said a new government with "new optimism" and "total conviction about the way forward" could find a way.
The EU has insisted repeatedly that it will not renegotiate the terms of the divorce, while parliament has voted against leaving without an accord.
On Wednesday, a cross-party group of lawmakers made a fresh bid to block such a scenario happening in future.
Finance Minister Philip Hammond warned Johnson and other leadership rivals not to make promises on Brexit they could not keep.
"The parliamentary arithmetic remains exactly the same, the European Union's position remains exactly the same," he said at a Bloomberg event.
Keep speaking directly
Johnson was a leading figure in the 2016 referendum vote to leave the EU and one of Britain's most recognisable politicians, known nationwide as just Boris.
But he is also one of its most divisive, accused of lying in the Brexit campaign and drawing criticism for derisory comments about women, ethnic minorities and gay people.
In his speech, Johnson skipped over an underwhelming two-year stint as foreign minister and instead drew on his eight years as mayor of London, which he said gave him the experience to govern.
Questioned about his comment that Muslim women in full face veils looked like "letter boxes", he said voters were alienated by politicians using "bureaucratic platitudes".
"I'm sorry of the offence that I have caused but I will continue to speak as directly as I can," he said.
However, he refused to clarify whether he had taken cocaine, an issue that has dogged leadership rival Michael Gove.
"The canonical account of this event when I was 19 has appeared many many times and I think what most people in this country really want us to focus on in this campaign is what we can do for them," he said.
Conservative lawmakers will hold the first of a series of secret ballots on Thursday on the leadership candidates, who also include Foreign Secretary Jeremy Hunt.
They will whittle down the field to two by June 20, and that pair will then be put to a ballot of around 160,000 Conservative party members.
The winner should be in Downing Street in late July but, if they cannot break the political deadlock in parliament, an election may be inevitable.
Johnson's supporters believe he is the only leader who can win against the twin threats of leftist Labour leader Jeremy Corbyn and eurosceptic Nigel Farage.
On Wednesday, he warned that his party faced "mortal retribution from the electorate" if it failed to deliver Brexit.
"Delay means defeat. Delay means Corbyn. Kick the can and we kick the bucket," he said.
But in a repeat of moves earlier this year, Labour led an effort Wednesday to try to take control of the Commons timetable to legislate against a "no deal" in the future.
The party's Brexit spokesman, Keir Starmer, said threats of crashing out of the EU were "disturbing, ludicrous and reckless".
Johnson has previously threatened to withhold the multi-billion-euro payment May agreed to cover Britain's EU liabilities.
But the European Commission warned Britain will still have to pay up even in a "no deal" scenario.