How can a white girl be accepted as a black girl?
It depends where you live. In certain parts of the US, it
depends on the density of black women per white women. Some black
women really don't want a white woman around and most certainly
don't like to see a black man dating a white woman. There is a
pride somewhere there. In other parts of the US, there are black
girls that will be friendly to white girls. In Canada we don't
really think twice about it. Most of us have black, Asian,
Indo-Canadians, etc., for friends and it's as simple as breathing.
If you like the person that's all that matters.
A black woman shouldn't be treated any different and try making friends with them the same way you would try to make friends with white girls in a new school or a new place of work. Take it slow and easy, then ask them out to a movie or lunch and gradually get to know each other.
Now, if you mean in terms of being accepted as a part of a different ethnic group than your own, there are things you can do if you really want to do so. You can choose to practice your accent and slang, choose clothes more likely to be worn by members of that ethnic group, wear more "ethnic" jewelry, listen to their music, adopt their philosophy on life, do your makeup and hair like them, etc. Several white women have done this, with interesting results. For one woman who tried this, more women of color asked her for help in stores and seemed more interested in striking up conversations, and she was approached less by men of color who prefer white women. However, there were some negative sides, like getting called ethnic slurs, but that is sure evidence of passing for another race or ethnic group. At any rate, there is no need to go to this extreme if you simply want to make Black friends. In that case, just treat them like gold.