As I played through Knights of the Old Republic 2, she tells us that she not only trained Revan, but also that she was also exiled and a Sith Lord. I’m wondering why she was exiled and what is her past that she doesn’t explain to Meetra. I know nothing of the old senile cynic before the Mandalorian and Jedi Civil War.
There’s not much information on her specific exploits (if indeed there were any), before she became Darth Traya.
What we do know is that Kreia was a former Jedi Master. Described as a "mystic and a historian" by the KotOR campaign guide, she was deeply contemplative and quite wise, fond of cryptic koan-like sayings. Her teachings were well-received by her students, but not popular with her colleagues.
She was "exiled" due having taught Revan, who later turned against the Jedi. The campaign guide for KotOR suggests that the proclivities of Kreia’s students were responsible for the Jedi Order’s condemnation of her.
When Kreia’s former students follow her one-time Padawan Revan to battle the Mandalorians, her rhetoric is denounced by Master Atris and the Jedi council.
Knights of the Old Republic Campaign Guide
Interestingly, this same source suggests that she “withdrew” from the Jedi Order before her fall, rather than being exiled, contrary to her own statements. Perhaps her censure and voluntary departure combined were enough to be perceived as exile in her mind— as Darth Traya, she is, after all, obsessed with betrayal.
It’s been speculated that Kreia and Arren Kae are the same person, despite the former referring to the latter in the third person, as mentioned in the Wookieepedia article for Arren Kae. If this is the case, then the reason for Kreia’s exile would be that she had a relationship and children, in contravention of the Jedi code.
One of Kreia's Padawans was a highly gifted Jedi who eventually became known as "Revan." Her teachings, however, were looked upon with concern by the Jedi Council. After Revan left Kreia's tutelage, he sought out other Masters to round out his education, apprenticing himself to Masters Zhar Lestin and Dorak. Eventually Revan came back to her to seek her guidance regarding the Jedi Council's stance of inaction against the Mandalorians. Revan then made up his mind to openly defy the Council. By the end of her time with the Jedi Order, Kreia's students were believed to be failures by Jedi Master Atris and the Council, because they followed Revan into the Galactic Republic's war against the Mandalorian Neo-Crusaders, and either fell to the dark side or abandoned their training. The Council denounced both Kreia and her teachings.