Lebedev attended St Barnabas and St Philip's Church of England Primary School in Kensington, followed by Holland Park comprehensive and Mill Hill boarding school. He then went on to study the history of art at Christie's in London. He has lived in the UK ever since, and became a British citizen (with dual nationality) in 2010.
On 25 March 2010, just weeks before it was due to close, Lebedev bought The Independent and The Independent on Sunday. On 26 October, the i newspaper was launched, the first national daily newspaper to be launched in the UK since The Independent in 1986, at a time of falling newspaper circulations and title closures worldwide. In 2011, he launched The Journalism Foundation, to promote "free and independent journalism throughout the world", although it was closed down after a year.
Lebedev is the patron of the Evening Standard's Dispossessed Fund, which helps to address poverty in London, and has raised over £13m since its launch in 2010. In 2018, he launched #AIDSFree, a cross-title campaign between The Independent and Evening Standard to raise money for the Elton John AIDS Foundation. In 2019, he announced that both newspapers' would launch a multiple-year campaign to tackle homelessness in London and around the world.