Hazelle worked as a washer woman. She would collect people's dirty laundry, wash it, and then return it to the owner.
This business did rather well, until right before the 75th Hunger Games. Gale had been caught poaching, and was beaten brutally. People stopped using her services for a bit after that, and her family was starving.
Afterwards, Katniss and Peeta hired her to take care of Haymitch. This results in extra money for her, and a much higher standard of living for Haymitch.
Hazelle was devoted to her family. Her husband died in a horrible mine accident, the same one that killed Katniss's father, leaving her alone at home with her three children. She was already pregnant with Posy. Gale, the oldest, helped take care of the family, bringing in food and money by poaching with Katniss Everdeen. Her sons Gale and Rory, also helped by applying for tessera when they turned twelve. Vick and Posy, her youngest children, were her reason for working so hard. It is known that Hazelle and the rest of her children escaped the District 12 bombing. They are seen in District 13. In fact, her youngest child, Posy, is the one to comfort Octavia after she is released from a prison with Katniss's other stylists.