There are several ways Welty characterizes Phoenix. Two of the more obvious are through external description and through the name given, which is symbolic. Look at the first paragraph:
It was December—a bright frozen day in the early morning.Far out in the country there was an old Negro woman with her head tied red rag, coming along a path through the pinewoods. Her name was Phoenix Jackson. She was very old and small and she walked slowly in the dark pine shadows, moving a little from side to side in her steps, with the balanced heaviness and lightness of a pendulum in a grand-father clock. She carried a thin, small cane made from an umbrella, and with this she kept tapping the frozen earth in front of her. This made a grave and persistent noise in the still air, that seemed meditative like the chirping of a solitary little bird."
As for her name, this symbolizes that she will fall and rise again, be burned and returned to life, like the mythic bird.