Ages 0–11 — The squib
Morrigan grows up in the mansion of her obscenely wealthy family in Hampshire and is a strange child from the start. She walks early and talks late and is thought mute for a long time. As it turns out she can hear the instructions of her governess, she just decides not to listen.
Her early childhood is somewhat uneventful. She likes to run away and pretends to live as a feral cat in the "forest" (her parents' apple orchard). But the massive gardens and high walls of her home allow her to indulge in such childish whims without putting herself in harm's way.
The girl does not show a lick of magic until she is ten and is thought of as a squib, even though she swears up and down that she is capable of great feats of magic, only she achieves them while sleeping, in her dreams. It is explained to her that dreams do not count if she wants to go to Hogwarts.
A few weeks before her 11th birthday she wakes up with two additional pairs of arms and one additional set of eyes on her forehead. She is quite satisfied with her new form, but her parents are horrified. In the waiting room at St. Mungo's, an owl with her Hogwarts Letter of Admission arrives.
The healers can undo the arms quickly, but it takes them weeks to dispel the magic that caused the eyes. Morrigan does attend the sorting ceremony with a bandage over her forehead, is sorted into Ravenclaw and has to sleep in the hospital wing for the first few weeks of her first term.

Ages 11–17 — Morg the hag

In Hogwarts, her magic abilities remain very inconsistent. She is capable of great effusions of magical power when she is emotionally riled up or really absorbed into the process, but struggles with fairly simple charms; It is like her system suffers from some sort of disturbance. Over the years she slowly manages to get into a routine, but most of her spellwork remains subpar.
She is very capable in theoretical subjects, very bright. Astronomy, Divination, Arithmancy, Runes, Potions and later Alchemy — these come easy to her. Charms and Transfiguration remain her weakest subjects.
She struggles through OWLs but stays in school. In sixth year her Ancient Studies Professor becomes convinced that the strange manifestations of magic that occasionally appear around her are signs that she is connected to ancient magic, possibly of celtic origin. With time, Morrigan becomes first amenable to this explanation, then obsessed with it. She dives into ancient studies and loses interest in everything else.

Socially, she is an outcast. The boys are not into her looks. She catches the nickname "the hag" and "Baba Yaga". Morg does not care one bit for boys' opinions and does not yield to anyone's expectations. She is too eccentric, queer and rich to care about her social prospects.
She makes NEWTs and takes off to travel the world.

Ages 18–30 — Wayward woman
At 18, Morrigan declares herself as "Britain's youngest spinster" to her family and tells them that she will travel the world and research ancient traditions of magic instead of coming out to society. And then, for twelve years, she does just that.
Her adventures are numerous and it would take up too much space to get into them. She travels from Egypt, to the Aztec ruins, to Russia, the Balkans, to Asia. She goes wide and far and learns a lot of things and forges many academic connections.

Ages 30–33 — Studies in Britain
When she returns to Britain, she continues her research and it soon points her to the most important stronghold of ancient magic that exists there today: Hogwarts.
So, she decides to return. Admittedly, educating children is not something she has much experience in, but she will learn on the job, somehow. Most importantly, she will have access to the castle.
Her former Ancient Studies teacher writes her a strong letter of recommendation and she has ample relevant life experiences, not the least of them her peculiar abilities.
She is admitted for 1894 to start teaching.