Mel was a computer programmer and fitness freak who first showed up in Terror of the Vervoids and gave the Sixth Doctor a massive heart attack by forcing him to exercise and drink carrot juice instead of ingesting glorious pies.
Her timeline is extremely confused. It makes River Song's timeline look positively normal. I'm going to try to explain it. But I haven't seen any of Trial of a Time Lord, or any other Six/Mel serial, so. . .
Sixie first encounters her in Trial of a Time Lord, when seeing her in the trial records, which contain his future. Vale's Yard tries to use this as evidence against him, saying that anyone who would bring a companion that incompetent and annoying is too irresponsible to travel through space. Six hasn't met Mel yet (remember, the video is from his future), so has no idea what the hell Vale's Yard is talking about. Mel is eventually dragged from her future adventures with the Sixth Doctor, to testify for "current" Six. So there first ever dialogue:
Mel: Finally a familiar face, Doc! First time you're the best dressed person in the room!
[Gallifreyan jury and prosecutor harrumph]
Sixie: Who in the Rassilon's posterior sphincter are you?
This was at least a better first meeting than Peri got.
Then, after Vale's Yard is cut down to size, Sixie later on takes on famous companion Grant Markham. But I don't have to tell you his story, as everyone already knows it.
The Sixth Doctor later on finally meets Mel for the first time, from her point of view. Together, they beat up giant plants, the Angel of Deepak Chopra, and the Vale's Yard himself yet again (and the Rani, but all characters are the Rani, so that's redundant). Then Six hits his head on the console, and becomes Seven. Then McCoy and Mel actually have a good rapport, but eventually Mel chooses to leave the Doctor in an underrated departure scene.
He unloaded Mel onto intergalactic pimp Sabalom Glitz who likely abandoned her after enslaving her into a brutal existence of sexual torture. Bravo Seven.