Carhillion of the Fold is a famed sorcerer from Melfia. He spent much time honing his sorceries and trying to “perfect his art” in one of Melfia’s renowned academies. While at the Academy he taught other students in the arts of sorcery. He believes there are no shortcuts to mastery, and was likely one of the scholars frustrated at Pyromancer Glocken and his Quartz Ring replicas.
Carhillion argued that the academies were terrible places to learn magic, which angered at least Pyromancer Glocken. Carhillion believed that the scholars and teachers there were not interested in “truth”. Rosabeth explains how he once spoke excitedly about Drangleic, believing it held remnants of great power, and that a deep abyss surrounded it. Carhillion himself elaborates on this further; the remnants he referred to are the forces of magic and souls that lie dormant in this land, perhaps meaning the spells that were developed in Drangleic and the unique souls that can be used to develop new spells.
What Carhillion seems most interested in is reaching the “limits” of sorcery. He speaks of crafting new spells, and the origins of sorcery. He says that sorcery is yet a mystery to him and that he wishes to mature in it with his pupils. What seems to fascinate him is a “dark power” that he senses in Drangleic, something akin to sorcery, only more primordial. It is clear that he is referring to hexes and the Dark. It may be that hexes are the “truth” that he is searching for, as they are spawned from the Dark Soul, humanity's common calling card.
Left to his own devises it seems that Carhillion would become fascinated with the Dark and take a similar path to Felkin the Hexer. However, we see no evidence of this in game, as Carhillion does not change from the moment we meet him.
While travelling to Drangleic he encountered Rosabeth of Melfia who was a student at his academy. They spoke once or twice, but never traveled together. When she disappeared he wondered what happened to her, but was too interested in his study of Drangleic to search. It is likely that Rosabeth does not have a high enough intelligence for Carhillion to take her on as a pupil as even when they are reunited she says that she has decided to "keep at it alone".
Carhillion’s membership in the “Fold” is interesting to note. It may have been the title that was given to him while he was a teacher at the Academy, which is where he was most famed. That would explain why he uses it to identify himself to the Player. The Fold is likely the administration at the Academy whose shared aim was teaching students sorcery and pyromancy, but since leaving the Academy, it seems strange that he continues to use the title as his aims and values no longer align.
TL:DR - Carhillion is a famed sorcerer from Melfia. He spent much time honing his sorceries in one of Melfia's renowned academies, but he argued that academies were terrible places to learn magic and set out for Drangleic, a land in which the forces of magic and souls lie dormant.