The project sees the addition of a two-storey extension for more education work and is the first major facelift for the building in 30 years.
It is being backed by the borough of Hammersmith and Fulham to the tune of £5 million alongside the council's own drive to improve the rundown area around its town hall.
The theatre's new landlord, MPKRS, which bought the site last year, is also redeveloping the shopping centre under the venue.
Council leader Nicholas Botterill said the combination of plans to breathe new life into King Street could be "the start of the re-birth of Hammersmith town centre".
Jessica Hepburn, the Lyric's executive director, said: "I'm a big believer that cultural regeneration can really change a community. We're delighted to be putting culture and young people at the heart of the regeneration of Hammersmith and Fulham."
Breaking ground has been the culmination of seven years of work and fundraising with the final £750,000 still to be found. The extension will include a film and TV studio and wardrobe and scenery workshops, while the existing building will get a new bar and café.
Supporters include the Greater London Authority, the Arts Council and the Department for Education, because of the theatre's plans to more than double the number of young people it works with to about 50,000.