The main difference between 'a' and 'the' is that for 'the' the listener knows which object you are talking about. But, for 'a' the listener does not know the object you are talking about.
So the correct sentence would be:
After the death of the emperor she became the empress dowager. (There is only one dowager, so the listener knows exactly what you are talking about.)
I just wrote a post about this that has some more explanations and some example sentences:
The articles a, an, the
I hope it helps! Thanks,