Until and unless can be used in many of the same places, but they have slightly different meanings.
Until refers to the time that something happens, and may be used to imply a chain of cause-and effect.
I won't make any money until I get a job. ➔ I will make some money when I get a job.
She didn't return until last week.
I'm not going to discuss this with you until you calm down.
Unless refers to a chain of cause-and-effect without reference to a specific time.
I won't make any money unless I get a job. ➔ I will make some money if I get a job.
You can't become a great musician unless you practice a lot.
I will go to the beach on Saturday, unless it rains.
Since chains of cause-and-effect always move forward in time, these two concepts sometimes overlap.
In your case, both versions of the sentence are correct and mean almost the same thing.
Until you do that one role that people recognise you for, you don't become an actor.
➔ You only become an actor when you do that one role.
Unless you do that one role that people recognise you for, you don't become an actor.
➔ You only become an actor if you do that one role.