Before The Outbreak
Little is known about Molly's life before the zombie outbreak. She was living in Savannah, Georgia in an upper-class neighborhood known as Crawford. She had a diabetic younger sister.
During The Outbreak
After the outbreak, Crawford quickly became a fortified city within Savannah that had a strict code of ethics and regulations, which included all children and sick be expelled from their community. Molly's sister attempted to hide her diabetic condition for fear of being expelled, with Molly exchanging sexual favors with Crawford's primary physician Dr. Logan for insulin. Eventually Logan told Molly that he could not longer supply her. Molly's sister was forced out of Crawford, likely killed by walkers. Disgusted and angry, Molly left Crawford as well.
Season 1, Episode 4: Around Every Corner
Following her exit from Crawford, Molly became a scavenger around the city of Savannah. She first encounters Lee Everett and his group of survivors while Lee, Kenny and Clementine are searching for a boat to flee Georgia. She informs them about the injustice occurring within Crawford (careful not to mention her history with the area), before the survivors were all attacked by Walkers. Molly was able to help rescue the visitors to her city, but not before getting separated from Lee.
Thankfully, Lee is reunited with Molly and the rest of his group of survivors, and better yet, they soon discover one of the few remaining boats in Savannah. Certain supplies are found necessary for the group's plan to escape, but the majority of the supplies are only available within Crawford. A party is formed to search the neighborhood, with Molly among the group. While in town, Lee can discover Molly's connection to the city as well as her past relationship with Dr. Logan.
Given certain circumstances, Molly can die during the Raid in Crawford (namely if the player doesn't bring Clementine AND doesn't rescue Molly in time.) Even if Molly is saved, she still decides to leave the group, deciding she's more cut out to be a loner anyway. It is unknown if Molly will return in Season 2.
Molly is depicted as being very resourceful and crafty. She uses typical mountaineering equipment to scale the buildings of Savannah, is constantly scavenging for anything of possible usefulness and has a complex system of using church bells to draw Walkers into a centralized area, leaving other areas clear to rummage. While her age is never disclosed, her appearance and mannerisms suggests she's young, either late teens or early 20s. Her greatest dedication was to her sister, given her acts of self-sacrifice and eventual rejection of Crawford. She can be very sarcastic but is generally moralistic in her views; she regards Crawford's actions as just as evil as the Walkers, and it is implied that she leaves the group as to not take up one of few precious spots on the boat. When she is first introduced, Lee and Kenny mistake her for a man, suggesting certain tomboy attributes.