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.
(..) You're angry with me. Of course I'm angry with you. Most of my life, you've either manipulated or humiliated me. You used me... to fulfill your ambition, while at the same time taking my wife as your mistress and forcing me to adopt her bastard son! Please, tell me... (..)2017 TvShow: Vikings Title: Revenge Season: 4 Episode 18