Not Alone
In the Odyssey, in fact, Kirke [Circe] does not live all by her lonesome on the island of Aiaia [Aeaea]. In Book 10, Odysseus says that Kirke's house is tended to by certain wood-nymphs, who "come from groves", and by a couple of varieties of water-nymphs, who "come from springs" and "from the sacred rivers flowing seawards". It is unspecified whether these nymphs are local to the island but it seems reasonable from the narration to understand them as Aiaia's indigenous inhabitants.
In Madeline Miller's Circe, after Helios banishes the title character to the island, the place eventually becomes a sort of juvenile delinquent rehabilitation centre for wayward nymphs, which then explains the origin of the goddess's attendants. The only other sentient lifeforms on the island are wild animals. Other ancient sources describe Kirke's situation, however, similarly to what we read in the Odyssey.
Like other goddesses, Kirke is waited upon by other lesser divinities. (Cf. the great Artemis, who is famed for haunting the mountain wildernesses of Arkadia [Arcadia] and Lakonia [Laconia], in which activity she is sometimes accompanied by sixty Oceanids, from another class of water-nymph.) According to Apollonius Rhodius' Argonautica 4, all of Kirke's housework is done by Naiads (river-nymphs), or at least such is the case in the generation before the Trojan War, when the divine witch's niece Medeia [Medea] visits her. In Ovid's Metamorphoses, upon Odysseus' arrival on Aiaia, even Nereids (from a group of powerful sea-goddesses) are among the nymphs tasked with sorting through Kirke's herbs and weaving materials.
Admittedly, Miller's novel actually does not contradict any of this, since these attendant nymphs are almost always incidental background extras in the scenes in which they appear in the ancient texts, unnamed and sans lines of dialogue, so it is in fact open to interpretation how exactly they might have gotten there. My interpretation is that they are simply minor deities filling their spots in the hierarchy under a greater deity, just as it is, for instance, in the entourages of Aphrodite and Dionysos [Dionysus].
In the Odyssey there is another goddess who is very similar to Kirke in terms of her relations with Odysseus, namely the sea-nymph Kalypso [Calypso], whose island of Ogygia is described in even more dramatic terms as a deserted place, very seldom visited by any other living beings, including the gods themselves. Upon his arrival there to deliver a message, the god Hermes is almost complaining while mentioning how he had to "traverse those endless briny waters, with not one town to be seen where human beings make sacrifice to the gods" in order to reach this destination.
But even here Kalypso has handmaids (presumably fellow nymphs) to serve her meals. (In Book 5 they seem to pop out of nowhere with dishes of nectar and ambrosia, never having been mentioned before; nor do they come up again after the meal scene herein.) Remarkably, in Fabulae 125, the Roman writer Hyginus says that it is Kalypso who lives on Aiaia Island! (With such anomalies, when it comes to Hyginus, however, one can generally assume that he is confusing one character or aspect of the story with another.)
Not Imprisoned
From what I could find, there is no ancient reference to any idea that Kirke is sequestered to or incarcerated on Aiaia. On the character-writing level, the reason she resides in her seclusion (relatively speaking) seems to be in order to increase the spookiness and mystique of this very powerful immortal witch. Within the story itself, as with Kalypso, although it is not explicitly mentioned, the reason for the seclusion appears to be simply because this is the manner in which she wants to live.
In Metamorphoses 14, Kirke meets and falls in love with Picus, the semidivine king of Ausonia on the Italian mainland, because she goes into the hills of Laurentum in search of herbs. Line 348 is somewhat ambiguous as to where Kirke is coming from on this occasion, saying that "she had left the Circaean fields, called so from her own name." Presumably this refers to a place on Aiaia. Depending on where one locates the otherwise mythical island, this might not be saying much, since, in Roman times, it seems to have been identified with a place which would have been within Picus's territory. According to A.S. Kline's Poetry in Translation website, Aiaia is not really a proper island but rather
the promontory of Circeii. (Cape Circeo between Anzio and Gaeta, on
the west coast of Italy, now part of the magnificent Parco Nazionale
del Circeo extending to Capo Portiere in the north, and providing a
reminder of the ancient Pontine Marshes before they were drained, rich
in wildfowl and varied tree species.) Cicero mentions that Circe was
worshipped religiously by the colonists at Circei.
At the beginning of the same Metamorphoses book, Kirke travels on foot across the sea, "dry shod", seemingly at an incredibly rapid pace, to get from her palace to the Strait of Messina where she proceeds to poison the waters which soon afterwards transform the beautiful sea-nymph Scylla into a humongous monster. If Kline's identification of Aiaia with Cape Circeo is correct then the goddess has basically walked roughly half the distance down the entire western coastline of southern Italy.
Westwards to Warmer Weather
Apollonius Rhodius' Argonautica appears to support this identification, having Kirke's brother Aietes [Aeëtes] locate her roughly in the same neighbourhood, albeit in Tyrrhenia, which was to be found farther northwards in Italy. In Book 3 Aietes tells the Argonauts that he accompanied his father Helios in his chariot when the Sun-Titan was taking Kirke to Tyrrhenia. No further context of the circumstances at the time is provided.
Valerius Flaccus' Argonautica suggests that Kirke, earlier in her life, lived with Aietes in the land of Colchis, over which country her brother ruled as king. In Book 7 Medeia reminisces of the time when her aunt was carried away from there "by winged serpents" or dragons, and apparently the implication is that these reptilian creatures were the steeds of Helios' chariot in the scene mentioned by Apollonius. This would mean that Kirke lived with Aietes at least until his daughter Medeia was old enough to remember her leaving.
In the same book, Aphrodite appears to Medeia disguised as Kirke, and Medeia interrogates her as to why she left, seemingly a long time previous to this encounter.
"What sojourning was more pleasing to you than my father’s land?"
This implies that, whatever the case, Kirke departed from Colchis of her own free will, or at least, as far as Medeia is aware, it was not under any duress. Aphrodite, still in disguise, responds with a few jibes about Colchis, calling it eternally cold and its inhabitants "unprofitable". She then goes on to boast about how she is now the consort of Picus and, by extension, so it seems, also queen of Ausonia.
If not for what we know of the interaction between Kirke and Picus in the Metamorphoses, we might not necessarily think to doubt what Aphrodite reports here about those two characters. Ovid tells the story of how Kirke tried to woo Picus and, being rebuffed by him, transformed him into a woodpecker. Nothing further is said there about her thereafter taking over his kingdom or anything of the like, an interest which she does not seem to have held.
Bad Tree, Bad Fruit
Wikipedia claims that:
Some say she was exiled to the solitary island of Aeaea by her
subjects and her father Helios for killing her husband, the prince of
Colchis.
To me this seems to be almost certainly confounding Medeia's story with that of Kirke. Medeia is famous for having killed her own brother, who would have been the prince of Colchis, while being pursued on her way out of the country by her father.
There does happen to be one ancient source, though, which fleshes out Kirke's "supervillain origin story," although this is a late euhemerisation mentioned by Diodorus Siculus in his Library of History, in which Aietes and Perses are sons of Helios who are made out to be cruel barbarian kings in Eastern Europe. Perses becomes the father of Hekate [Hecate], who is not a Titan goddess here at all but rather a human princess who grows up to be an even viler individual than her father. After assassinating Perses using poison she succeeds him on his throne and founds the temple of Artemis Tauropolos, "commanding that strangers who landed there should be sacrificed to the goddess".
After that, Hekate marries her uncle Aietes and bears him children, among whom are Kirke and Medeia, the former of whom, being just as rotten as her parents, proves that the apple doesn't fall far from the tree, but the latter of whom, rather, disproves this saying, becoming an advocate for practising proper hospitality towards strangers rather than killing them for sacrifice.
From her mother as well as her own studies, Kirke becomes most proficient in the knowledge and use of drugs (i.e. herbs for sorcery). According to the Library of History 4.45.4-5, Kirke is then
given in marriage to the king of the Sarmatai [Sarmatians], whom some
call Skythoi [Scythians], and first she poisoned her husband and after
that, succeeding to the throne, she committed many cruel and violent
acts against her subjects.
For this reason she was deposed from her throne and, according to some
writers of myths, fled to Okeanos [Oceanus], where she seized a desert
island, and there established herself with the women who had fled with
her, though according to some historians she left Pontos [Pontus] and
settled in Italy on a promontory which to this day bears after her the
name Kirkaion [Circaeum].
Neither in this version nor in any other is anything explicitly said about any relative of Kirke (whether father, grandfather or uncle) exiling her anywhere. In fact in the mainstream version of the mythology, in which she is undoubtedly an immortal goddess, her presence on Aiaia seems to be as much a gift to her from her father as it is for Helios to give Colchis to Aietes. (Granted that the latter of these is told to us in generally less ambiguous terms, with Helios showering Aietes with a variety of divine presents conveyed from other gods after he has settled in Colchis.)