What interests me the most is understanding the context in which the Bible was compiled and edited. I've already read some books like Brettler's How to Read the Bible and went through all of Christine Hayes lectures on youtube, but I'm looking for something that focuses less on the text and more on the historical setting.
My go-to for the historical context of Scripture is Lester Grabbe's Ancient Israel : what do we know and how do we know it?.
It goes through pretty much the entire history of ancient Israel and discusses all the Biblical stories, outlines various extrabiblical texts and discusses the archaeological evidence that pertains to the situation. It is an absolutely essential book for studying this part of history. I'm not at home at the moment, so I'm not entirely sure exactly how much it details the Exile and the Babylonian Captivity, but it is extensive over the rest of Israelite history.
Grabbe also has a book specifically about Exile period Judaism, so I will point you to his book as well: Grabbe, Lester L., "A history of the Jews and Judaism in the Second Temple Period", vol.1 (T&T Clark International, 2004).
I have not read that second book, so I can't personally vouch for it, but if Grabbe's first book is anything to go by, it will be scholarly and rigorous and very informative.
A couple of books edited by Oded Lipschitz (yes, real name, stop laughing) and Manfred Oeming should also be pretty good:
Again, I haven't read these books specifically, as I haven't specialized in this particular time period, but Lipschitz is a well-respected scholar in the field, and therefore I would certainly go to these works were I researching this period. A number of the articles are written by either well-respected scholars (such as famed epigrapher Andre Lemaire, Lester Grabbe, David Ussishkin, etc.) or people I know personally and are very good scholars.
These will most likely give you all the background you are looking for and more besides. I'm not as up to speed on the textual compiling so I hesitate to offer recommendations on that angle.
As far as history is concerned, you want to look at Amelie Kuhrt's two volume history of the ancient near East. From there, I would use her bibliography to find more focused sources than her broad treatment.
Re: Brettler--remember that he's not the be all/end all authority on textual composition/redaction. A great start for sure, but you'll also want to check out Carr, Schniedewind, Baden, and a bunch of others to get the full picture of theories about these issues.