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Vladimir Putin is engaging in "barbaric and indiscriminate" tactics targeting civilians, Boris Johnson has said.
The prime minister said the Russian president had "fatally underestimated" the willingness of the Ukrainian people to fight and the resolve of the West.
Speaking during a visit to Poland, he also said the UK had set aside £220m in humanitarian aid.
Meanwhile, Foreign Secretary Liz Truss has said "nothing and no one is off the table" for further sanctions.
The UK is considering calling for Russia to be expelled from the UN Security Council, the BBC understands.
Mr Johnson's comments came in a press conference with his Polish counterpart Mateusz Morawiecki, where Mr Johnson said he was "increasingly confident" Russia's invasion would fail, adding that Mr Putin "must fail" in his aims in Ukraine.
He said Mr Putin was prepared to "bomb tower blocks, to send missiles into tower blocks, to kill children, as we are seeing in increasing numbers".
The Russian army continues its advance on Kyiv, with satellite images spotting an armoured convoy about 40 miles long.
Air-raid sirens were sounded again in the capital overnight, and there were also reports of fierce shelling in other cities including Chernihiv in the north.
The fighting escalated on Monday, with missiles killing dozens of civilians in Kharkiv, Ukraine's second city. Ukrainian president Volodymyr Zelensky has described the bombardment as a war crime.
Mr Zelensky said Russia was committing a "terror against Ukraine" by bombing the city's Freedom Square and civilian populations, adding it was the "price of freedom" for his nation.
In a speech after the press conference, Mr Johnson announced the emergency aid and said the UK had placed 1,000 troops on standby to help with the humanitarian response in Europe.
He said the government was making it easier for Ukrainians in the UK to "bring their relatives our country", adding this could end up being more than 200,000 people as part of an extension to the scheme to help those fleeing the war.
The PM's spokesman said the scheme would be extended to include adult parents, grandparents, children over 18 and siblings of those settled in the UK.
Home Secretary Priti Patel is expected to announce further details of the extension, as well as more on a sponsorship scheme for Ukrainians who do not have family in the UK in the House of Commons.
Mr Johnson also warned there would be a cost to Western economies as a result of sanctions, adding it was essential Europe "wean ourselves off dependence" on Russian oil and gas exports.
Following his speech, Mr Johnson was confronted by a Ukrainian who criticised Nato's response to the invasion, and asked why Russian oligarch and Chelsea Football Club owner Roman Abramovich, who has close ties to Mr Putin, had not been yet sanctioned by the UK.
Daria Kaleniuk, the executive director of the Ukrainian civil society organization Anti-Corruption Action Centre, told the PM: "You are talking about more sanctions prime minister but Roman Abramovich is not sanctioned, he's in London, his children are not in the bombardments, his children are there in London."
Meanwhile, Foreign Secretary Liz Truss has warned "nothing and no-one is off the table" for further sanctions on the Russian economy, Mr Putin and the "highest echelons of the Russian elite".
She told the UN Human Rights Council in Geneva the Russian president was responsible for civilian deaths and more than half a million refugees so far.
Ms Truss said: "The blood is on Putin's hands, not just of innocent Ukrainians but the men he has sent to die.
"We're using our collective heft, making up over half the world's economy to cut funding from Putin's war machine and we're delivering severe economic costs through these sanctions as ordinary Russians are finding from queues at their local banks and rising interest rates."
In other developments:
Mr Johnson will later meet Estonian leaders and Nato secretary-general Jens Stoltenberg and is due to discuss the response to the unfolding humanitarian crisis on Ukraine's border, as well as European security.
He will also speak to British troops serving in Estonia - which shares a border with Russia.
Meanwhile, Deputy Prime Minister Dominic Raab has warned that Russian President Vladimir Putin could retaliate with increasingly "heavy-handed tactics" if his invasion of Ukraine faces further hitches.
He told BBC Radio 4's Today programme: "We can expect, for every stutter and stumble, him to try and come back for even more heavy-handed tactics, but that is a sign that the initial phase at least - and this is going to be a long haul - has not lived up to his expectations."
Both he and the prime minister said on Tuesday the Russian president and military commanders could face charges for war crimes if reports of attacks on civilians could be verified.
In his speech, Mr Johnson said: "I think that everybody involved in the Russian onslaught should understand that all this will be collated in evidence to be used at a future time in what could be proceedings before the International Criminal Court."
Shadow defence secretary John Healey said the Labour party would support the UK government in responding to the invasion, adding: "The most important thing is there is a united UK voice, standing with the Ukrainians."
The Prince of Wales has also condemned the "brutal aggression" against Ukraine, in the strongest comments so far from the Royal Family about Russia's invasion.
