来る is most often read as
くる. However, it can also be read as
きたる mostly an archaic reading? Does it carry a different, or any additional meanings?? When are "appropriate" (socially, grammatically, etc.) times to use it?
Basically, it is literary (and archaic), but there are some situations when it is still appropriate to use it in a modern context.
Outside of these usages, it has a literary flavor, sort of like "is come" or "cometh" (rather than "has come") in modern English.
Incidentally, I learned that although it looks like 来る + past/perfect aux. たる, it is actually a separate word derived from 来る + 至る; I just checked the 広辞苑 and 日本国語大辞典, and they agree with this (although obviously, sometimes the word きたる will in fact be 来る + たる). Special bonus trivia: the 来る + 至る /kitaru/ got its big break in the world of kanbun 漢文, as a pinch hitter for 来(る). (And it is true that the character 来 is generally rendered /kitaru/ in kanbun.) I can't find a reliable source for why kanbun kundoku doesn't just use /ku(ru)/, but some (e.g. fontomanie at Chiebukuro) suggest that it was because /ku(ru)/ didn't have enough syllables for convenient okurigana affixation (cf /nasu/ instead of /su(ru)/ for 為).
So, the specific literary flavor is that of kanbun or heavily Chinese-influenced Japanese, not "pure" classical Japanese.