Harli's parents were regular mages who inherited her grandfather's research results after he discarded his humanity. Mages wanted to steal said researched result, so the Borzaks were treated as herectics for no good reason and had everything taken away from them, except Harli's life. Since then, Harli has been raised by her teacher this entire time.
A few years before the main plot, she talked about Francesca's Holy Grail War project to her teacher. The teacher don't seriously try to stop her from taking the invitation, but very strongly advises her against, giving words of warning like (note: this is all paraphrased, partly from memory, not real translations)
-"No action is without consequence. If you destroy the Mage's Association, you would be making every mage out there your enemy."
-"I saw a Heroic Spirit once just a couple months ago when I was in a Rail Zeppelin auction, and those are some truly scary familiars. Definitely not recommendable as a plaything for your grudges."
-"You had the chance to escape with life and Magic Crest, so it would be very thoughtless of you to waste it by becoming that demonic Francesca's pawn. That Lord El-Melloi II I saw in the Rail Zeppelin holds a classroom that accepts people contrary to the magecraft world like you, so I think that would be a good place for you to move past your trauma."
But despite the whole conversation, Harli insists she's already past the breaking point and takes Francesca's offer anyway.
The teacher is described as an 80+ sly granny with the appearance of a classy little girl with an elegant dress and a crow mounted on her shoulder. If you checked the Rail Zeppelin background characters, it checks out.
The Rail Zeppelin anime is very consistent with its background characters, actually. You can see Harli's teacher a couple more times.