This play takes place in the early Fifties, which was a time of enormous African-American migration from the South to the North. Black people moved to Chicago, Detroit, Pittsburgh, and other cities where there was less overt discrimination and more job opportunities. The South was still not all that industrialized and manufacturing of all sorts offered African-Americans employment.
There is nothing in the play that I have ever noticed to suggest why Berniece ends up in Pittsburgh, but it did have a thriving black community at the time, and it often happened that a friend of a friend or a friend of the family would help a new person set down roots in a new place. People often came North and stayed with someone they knew first or second hand until they found work.
The fact that Crawley was killed might have played some part in Berneice's decision to get out of town, too. That, plus the promise of greater opportunity in the North, and perhaps her hope that leaving the South would provide a better climate in which to raise her daughter all no doubt would have played some part in the decision of a woman of her time, place, and color.