Introduction
Samuel Sandmel writes on the period:
“The signs of the postexilic period are unmistakable. Israel is spoken of not as usual as an ‘am, a nation, a people, but rather as a qahal, a congregation. The entity is a reduced community, no longer extending from Dan in the north to Beer Sheba in the south. Ammonite and Moabite may not join it, but Edomites and Egyptians may; we are past the time of the first missionary movement. The religious purity of the congregation is to be maintained; over and over again there occurs the formula, ‘You shall cleanse the evil from the midst of your people.’ Evil things, ‘abominations,’ are not to be tolerated; indeed, to abominate (Hebrew ta’ab) takes on the meaning, ‘to exclude from the congregation.’ The attitude towards heathens is most severe, and the worst of all transgressions is apostasy from Yahve. Yet on the other hand, an extensive humanitarianism is to be found in many of the laws and much of the exhortation.” (The Hebrew Scriptures, pp. 414-415)

Richard D. Nelson writes on the composition:
“The original book of Deuteronomy seems to have consisted of the reform-oriented law code proper (chaps. 12-26), framed by an introductory exhortation (chaps. 5-11) and some concluding chapters that directed the law at the readers of the book (chaps. 27-30). There is no agreement whether an even earlier and shorter form of the book once existed. A puzzle feature of Deuteronomy is its alteration between second-person singular and plural address. The plural portions often seem to be somewhat later than the singular portions, but there is no completely satisfactory explanation for this phenomenon.” (Harper’s Bible Commentary, p. 209)

Jay G. Williams writes:
“It is this emphasis [on one central shrine], in particular, which has led scholars to identify Deuteronomy as the scroll of the law found in the Temple during the reign of King Josiah in the seventh century. According to II Kings 22-23 this scroll led Josiah to initiate a reform of the religion of Judah which, in particular, involved the destruction of all places of sacrifice except the Temple in Jerusalem. Since only Deuteronomy, of all the books of the Torah, calls for such a reform and since it is inconceivable that such an important book of the law would have been lost after Josiah’s time, it is likely that the identification of Deuteronomy as the discovered scroll is correct. The fact that Deuteronomy often reflects both the language and the thought of the eighth century prophets helps to confirm this identification.” (Understanding the Old Testament, p. 137)

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Authorship 📜
The general consensus among scholars is that no one person–Moses or anyone else–wrote the Torah/Pentateuch/Five Books of Moses by himself. Instead, various authors, writing at various times BCE and with different sources and agendas, wrote parts of what became the Torah, and these sources were later combined, which resulted in some inconsistencies and contradictions. For a general overview, read the introduction to Richard Elliott Friedman’s book The Bible with Sources Revealed (https://biblebrisket.files.wordpress.com/2014/01/biblewithsources-intro-and-first-chapter.pdf). One of the most famous stories which demonstrates the use of sources is the narrative of the Great Flood. See the article here for more (https://isthatinthebible.wordpress.com/2016/11/06/reading-the-fractures-of-genesis-noahs-flood/).

The story of the discovery of the book in the 2 Kings 22:1-20 does not give any description of the discovered scroll, apart from an indication that it contained a number of curses that would be inflicted on Judah for its failure to follow its prescriptions. The reforms that Josiah instituted upon receiving the book, however, closely follow the instructions that are found in Deuteronomy — in particular the section of the book that is found in chapters 12 through 26, which is followed by a list of curses for those who fail the keep the laws and blessings for those who keep them. This has led to the conclusion that the book that was found was either Deuteronomy or this central portion of the book.
One explanation, for example, is that perhaps the contents of the book were compiled in an attempt to preserve information from an oral tradition when it was in danger of being lost. The precipitating event for this project might be the destruction of the Northern Kingdom of Israel in 720 BCE. There would have been a fear that the traditions surrounding the worship of Yahweh that had been maintained in Israel up until that time might be lost forever, so perhaps some enterprising individual in the surviving Kingdom of Judah undertook to write these traditions down. The resulting scrolls might have then been hidden during the reigns of Manasseh and Amon when the worship of Yahweh appears to have fallen into disfavor.

There is strong differences between the book of Deuteronomy and the rest of the other 4 books, primarily with their sources. The vocabulary is different, and there are different recurring expressions and favorite phrases. There are blatant contradictions as well.

Anyways, Deuteronomy has multiple authors, probably an addition one (D).

In 1943, a German biblical scholar, Martin Noth, showed that there was a strong unity between Deuteronomy and these six books of the Early Prophets. The language of Deuteronomy and parts of these other books was too similar for coincidence. Noth showed that this was not a loose collection of writings, but rather a thoughtfully arranged work. It told a continuous story, a flowing account of the history of the people of Israel in their land. It was not by one author. It contained various sections, written by various people (such as the Court History of David, and the stories of Samuel). The finished product, nonetheless, was the work of one person. That person was both a writer and an editor. He (the person was male, as we shall see) selected the stories and other texts that he wanted to use from sources available to him. He arranged the texts, shortening or adding to them. He inserted occasional comments of his own. And he wrote introductory sections which he set near the beginning of the work. Overall, he constructed a history that extended from Moses to the destruction of the kingdom of Judah by the Babylonians.

Dating

More on authorship:

Style