The origins of the Beta Israel are actually quite uncertain.

Up until the 1970s, the most frequent explanations posited the migration of a cohesive Jewish community into Ethiopia. The older of these interpretations drew upon the legend of the Lost Tribes of Israel, and argued that the Beta Israel descended from one of these missing groups.

A second interpretation was that the Beta Israel represented the migration of a Southern Arabian (Yemeni) Jewish community into Aksum/Ethiopia.

A third interpretation speculated about the possibility of a Jewish garrison existing at Elephantine, on the frontier of Egypt in the 7th-5th century BC, and the Beta Israel being descendants of this garrison that had migrated to Aksum/Ethiopia.

Since the 1970s, scholarly consensus has turned away from these migration theories.

After the 1970s, the scholarly consensus has been that it is impossible to identify a historical Jewish community that was socially and culturally distinct from surrounding Christian communities before the 14th century AD.

Certainly, there had been historical Judaic influences on ancient Aksum and medieval Ethiopia, for instance the contact and conflict with the Himyarite kingdom led by the Jewish king Dhu Nuwas. However, after the Aksumite conversion to Christianity in the 4th century, scholars speak of communities expressing a spectrum of belief, placing different emphasis on Judaic or Christian elements.

There are mentions of early medieval persons such as Eldad ha-Dani, Benjamin of Tudela and Queen Judith that are sometimes invoked to support the idea of a Jewish community within late-classical Aksum or early-medieval Ethiopia. However, there is considerable doubt about whether these accounts actually refer to the horn of Africa (in the case of the first two) and whether they represent a genuine account of a Judaic community in the region (in the case of the latter)

To the Beta Israel as a defined community, the earliest textual mention we have of them is in the 14th century, during the reign of emperor Amda Seyon. That emperor was a dedicated proselytizer of Christianity in the Ethiopian borderlands, and accounts of his reign mention his warring against people "like Jews".

After the wars of Amda Seyon, the period from the 14th to the 16th century can be characterized by a conflict between the Beta Israel and the Ethiopian monarchy. During this roughly 300 year period, there appears to be a growing religious-cultural distinction between the Beta Israel and their Christian neighbors.

Also, is there any historical evidence to support ties to the Kingdom of Israel under Solomon?

The legend of the royal family of Ethiopia being descended from Solomon and the Queen of Sheba is very likely a piece of political propaganda from the early 14th century, a few generations after the rise to power of the so-called "Solomonic dynasty"

I am going to shamelessly link to this earlier answer that explains the circumstances of the Solomonic legend.

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My information about the origins of the Beta Israel in Ethiopa was lifted pretty much entirely from the Encyclopaedia Aethiopica vol 1- A-C, particularly the entry on the Beta Israel, pp 552-558.