Mel is a computer genius from the near future (when her stories aired, that is. She’s from the early 21st century). As the Doctor, and the audience, first meet her after she’s already met him, we don’t find out much about her backstory or her family and friends.
The Seventh Doctor begins his tenure as a rather comical, light-hearted figure. Sylvester McCoy was known for his vaudeville-style clowning and humor and the early Seventh Doctor stories play into this. As his stories progress, however, he becomes a much darker, more still figure and he’s perhaps the most manipulative and deceptive of the Doctors. He has an incredibly close bond with Ace, who is very much his protégé and a daughter figure, and this relationship is a clear precursor to the more intense, more closely-examined Doctor/Companion relationships to come in NuWho (minus the romance). It’s highly implied during this time that the Doctor has a far more significant, secret past than his “Time Lord on the run” cover story, but this is quickly backed away from later. This ominous, looming secrecy gives the Seventh Doctor an air of foreboding and drama that counterbalances his early and occasional silliness quite well.
Dorothy Gale McShane, nicknamed Ace (which she says rather frequently) is troubled teenaged chemistry whiz, Ace enjoys cooking up bombs of her own recipe, Nitro-9, which she’s fond of carrying around just in case. While she was experimenting with explosives in her room, a time storm swept her off from contemporary (to when her stories aired) England to a spaceport in the far future, where she encounters the Doctor, who takes an interest in the displaced teen.