I found a rather extensive and well referenced list of Old Testament historical figures in this article. It lists 61 people, most of whom are identified in the records or monuments of non-Hebrew civilizations, or on archeological finds such as seals and signet rings. Included in the commentary before the list are three criteria for inclusion:
1) The name of the person must be found in some independent source.
2) The name must have a clear connection to a Biblical story. (A seal with the name Baruch may not be connected since many Baruchs may have existed, but "Baruch son of Neriah" found both in the Bible and and on a seal would likely be connected.)
3) The context (especially in chronology) of the evidence must fit with the context of the Biblical story.
The article further argues:
To my mind, it is not the verification of the major personalities that is so impressive but rather the verification of the more obscure Biblical characters is the more astonishing.
It could be argued that many of the finds suffer from selection bias (if you can find a connection to the Bible, you are more likely to publish a find than not). But that does not make the evidence go away, it merely lowers our expectation that new evidence will crop up in the future.
Whether these Biblical characters are consistent with outside history is hard to know since the independent evidence mostly records their names and positions. Most historical records from the time are uninterested in describing the character of historical figures. At most, we receive broad accounts of their actions and those do seem consistent with the Biblical sources, for what it's worth.
There is also evidence of New Testament characters to be found in Roman and Jewish records. I didn't find a list as condensed as the one for the Hebrew Bible, but this site includes not just independent sources for Biblical people, but also the political, social, intellectual, and cultural climate in which early Christianity was formed. If the list were compiled it would be fairly long, but I'd like to point out the unusual case of Gamaliel, who is well-regarded in both Jewish and Christian traditions.
We are quite dependent on Josephus for most of our history surrounding Judea leading up to and through the destruction of the Temple. He has gotten a bad name in recent years because his mention of Jesus (the Testimonium Flavianum) was "punched up" by some later scribe to be more favorable to Christianity. A 1971 discovery of an Arabic copy of Antiquities does not include as many Christian interpolations suggests the passage is not entirely spurious. Peter Kirby has written a pretty good survey of research surrounding the passage. On the other hand, references to other Biblical characters such as James, the brother of Jesus, and John the baptist are more commonly accepted. Of course, references to various officials who also appear in the Bible are not contested at all.
Jesus himself is of course well-attested in Christian sources, but also receives mention in non-Biblical texts. Not surprisingly given the antipathy of almost everyone to the early Christians, the non-Christian sources sharply contradict the Biblical account of his character and the meaning of his life.
A list of New Testament figures associated with archeological finds that match the above criteria include:
Lysanius - ruler of Abilene
Pontius Pilate - governor of Judea
Sergius Paulus - governor of Cyprus
Erastus - city treasurer of Corinth
Gallio - governor of Corinth
All of these are non-Jewish officials who left monuments recording their names and titles that fit the Biblical accounts. It also seems that Luke, the author of the earliest church history, had good knowledge of the rather intricate use of official titles in that period. None of this should be surprising, however, since all canonical New Testament texts were written within 70 years of the events they describe. It seems unlikely that the authors would need to (or even be able to pull off) inventing characters from whole cloth to fill out the stories.
More difficult is finding compelling archeological evidence for Jewish personalities—partially because Jerusalem was fairly effectively destroyed by the Romans in 70 AD and partially because most of the Jewish characters mentioned in the New Testament were unremarkable except for their connection to Christianity. (It seems unlikely that we would have any independent record of a Galilean fisherman, for instance.) Even so, a number of ossuaries (bone boxes) have been found with New Testament names. Few of them can be connected to a Biblical character according to criterion #2 above and those that have are in dispute according to criterion #3.
Some of the physical artifacts that seem to corroborate biblical figures appeared without provenance on the antiquities market. This can be a problem for historians. For instance, the second of two bullae attributed to Baruch ben Neriah is likely a modern forgery. Unfortunately, objects that connect to characters in the Bible command higher prices than other artifacts and so liable to be forged. (To be clear, forgeries fail the third criterion above.)
The solution for buyers (whether paying to own the object or investing credence in it) is to demand documentation showing where the object was discovered. For instance, several other seals similar to the Baruch bulla were found in an archaeological excavation in Jerusalem. We can be more confident that the object existed at a particular time that can be reliably estimated from context.