Martha Wayne was the wife of Dr. Thomas Wayne and the mother of his son, Bruce. When she and her husband were murdered by Joe Chill, they inspired Bruce to pursue a life of crime fighting. During the events of the series, it was revealed that she and Thomas had ties to Carmine Falcone and corrupt Mayor Hamilton Hill.
At some point, Martha met and married Thomas Wayne, a prominent doctor in Gotham City and humanitarian. However, in reality, he was a partner in crime and ally of criminal Carmine Falcone and corrupt mayor Hamilton Hill, who ran most organized crime within the city. During their marriage, Martha gave birth to a son, whom they called Bruce. Martha is said to have been aware that Thomas was a partner in crime and ally of both Carmine Falcone and Hamilton Hill. However, it is unclear how much she was involved in their operations and whether she herself was linked with their activities.
After learning of the trio's usage of Arkham Asylum to get rid of opposition, Martha became disgusted at her husband and the people he worked with. Upon learning this, she set out to expose him and his allies. It is unclear whether there was any other factors that motivated her to do so. It's possible Martha also asked the family's butler, Alfred Pennyworth, to help her expose Thomas.
Eventually, Hamilton Hill learnt about Martha's plans and set out to silence her. He hired Joe Chill, one of Falcone's hitmen, to assassinate them and make it look like a botched robbery. As the family returned from a showing of the Mark of Zorro at the Monarch Theater, Chill ambushed them in a nearby alley and held them at gunpoint.
As Thomas begged for their lives, Chill shot him twice in the torso, claiming it was nothing personal. He then shot him once through the eye, killing him. As Martha scream her husband' name, Chill shot her in the chest. As she bled to death, she called out for Bruce and tried to hold his hand. After Martha died, Chill took her pearls and set the gun on Bruce, but either fled before the police could arrive or chose not to murder the child.
Following Thomas and Martha's deaths, Gotham's elite began banding together to prevent them from suffering a similar fate. Chill was later caught by the police and sentenced to Blackgate, but was later murdered by someone on either Falcone or Hill's payroll. Bruce would later become the vigilante known as "Batman", hoping to avenge their deaths and right the wrongs in Gotham.
When allegations of his father's dealings became public, Bruce found himself unable to see either of his parents in a positive light. However, after learning about his mothers wishes to stop his father's crimes, it's possible he was able to forgive her.
According to Falcone, Martha had the "human touch" about her. This implies that she was very caring and considerate of other people. This may have also been the reason why Thomas married her. Unlike her husband, Martha was unwilling to turn a blind eye to the people who suffered and was willing to lower her reputation and social standing to help others. It was ultimately this that lead to both and her husband's deaths.
Martha was married to Thomas roughly nine or years before their assassination. It is unclear how much she loved him, but Falcone claimed that she was the only one who could control his temper. However, after learning about his criminal dealings, Martha planned to reveal his dealings to the authorities and expose his crimes, meaning she was willing to put her relationship with her husband at risk to protect others.
It seems Martha had a positive relationship with Bruce before her death, as Bruce recalls her being so kind to him as a child. Alfred implies that she had dealings with the managers at the Monarch Theatre to play his favorite movie whenever he wanted to see it. It's possible Martha wished for her son to turn out differently than her husband. After Martha's death, Bruce swore to avenge both her and Thomas' by ridding crime as the Batman. It's possible that, after learning the truth about his father and her plans, Bruce swore to carry on the role in her name.
The precise relationship between Martha and Alfred is unknown, but it's likely the two would speak about their thoughts about Thomas and his dealings. Alfred was aware of her plans to expose her husband and would probably of helped her. However, rather than speak up about it, he chose to leave their employment, but only looked after Bruce following their deaths. It's possible that Alfred would have looked after Bruce not only to make sure he turned out differently than his father but in memory of Martha.
The precise relationship between Carmine Falcone and Martha is unknown. However, it was likely that the two had little to no communication between them outside of meetings with her husband. It is implied that they had an affair, as Falcone could remember her "human touch". Although, this is probably very unlikely, due to Falcone's dealings and Martha's disgust for his and her husband's actions.
Like Falcone, the precise relationship between Martha and Hill is unknown. However, it was likely that they had little to no communication between them outside of meetings with her husband. However, after Hill learnt about her plans, he arranged their murder, meaning he was willing to put his greed above her and Thomas' lives. Hill may have also asked Chill to kill Thomas, knowing that killing Martha would have had repercussions from him if he was still alive.