She finally achieved/realized/fulfilled her 'life's ambition' [=the thing she most wanted to do in her life] when she started her own business.
He has 'ambitions' for an acting career. = He has 'ambitions' to become an actor. = His 'ambition' is to be/become an actor.
With her talent and fierce 'ambition' [=her very strong desire to succeed], she became a very successful actress.
(..) I understood. Sir... Sir... They say that they don't know any foreign language. The General has made out of this a personal ambition. Bologa, welcome. I caught them last night. We can't understand each other... And my french doesn't help... But Mr lieutenant says that they don't know french. (..)1965 Forest of the Hanged