This is quite a long chapter (81 verses) and it is also quite diverse in the topics it covers, which makes it a bit hard to find a unifying theme. As far as I can tell, this chapter is meant to give a broad overview of the Levitical tribe in every way that the Chronicler cares about them. It begins by listing all of the high priests from Levi down to Jehozadak and the Babylonian exile (which incidentally shows that Chronicles must have been written some time after the exile). Then it lists the clans of Levi (a traditional genealogy), and then it lists the genealogies of several temple musicians appointed by David, and then it concludes with the Levitical towns in Israel. The emphasis on the priesthood and temple musicians is relevant to the later narrative sections in Chronicles. The author of Chronicles shows a deep interest in everything related to the temple. It’s not clear to me why the author is interested in the Levitical towns.
In general, the length of this section is evidence of the author’s preoccupation with Levi compared to the other tribes like Reuben, Gad or Manasseh. All put together, those 2.5 tribes were describe in 26 verses, less than 1/3rd of the length of the passage for Levi alone. I think this preoccupation is driven by two factors: the first, as I said, is that the author is trying to glorify and honor the temple-building operation by David and Solomon, and the second is to implicitly legitimize the religious institutions in his own time. There are several tribes in the post-exilic period that maintain a lot of significance and power, and all of these tribes receive extended treatment in the genealogy. Foremost amongst them are Judah and Levi, with somewhat lesser coverage for Benjamin and I’d also say Simeon.
Those are my general thoughts. Now let’s dig into some details.
One of the first thing I noticed in v. 1-3 is that according to this genealogy Moses is just three generations removed from Levi. It is exceedingly unlikely that 430 years (Ex 12:40) could have passed in three generations, which indicates that genealogical telescoping is likely occurring here. That is, I think the Chronicler is possibly omitting people from the genealogical record for the sake of poetic symbolism or (more likely) because his original sources also omit certain people between Levi and Moses from the genealogical records.
Next, when listing the high priests in v. 3-15, Nadab, Abihu and Ithamar are all mentioned as sons of Aaron, but Nadab and Abihu died when they rebelled against Moses so even though they were both older than Eleazar, they never served as high priest and their families were also wiped out. Strangely, there are known high priests from Samuel and Kings, who are recorded by name in those books, but are omitted from this listing of high priests. For instance, Jehoiada (2 Kings 12:2), Uriah (2 Kings 16:10), Eli (1 Samuel 1:9) and Abiathar (2 Samuel 8:17) are all omitted. It's possible that Eli and Abiathar are left out because they weren't part of the priestly line that continued through Zadok, but I don't know why Jehoiada and Uriah are left out. The author of Chronicles demonstrates clear knowledge of Samuel and Kings, so these omissions cannot possibly be by accident or ignorance. It's definitely intentional, perhaps for brevity or perhaps because he's trying to emphasize the priestly line of Zadok vs. the other priestly lines (like Abiathar). Verses 50-53 gives the same priestly line from Aaron to Ahimaaz, because this is the line that is permitted to offer sacrifices in the most holy place (v. 49). It's pretty clear from here and elsewhere (especially 1 Kings 1-2) that there was a power struggle between Zadok and Abiathar, and the Chronicler emphatically sides with Zadok.
This section of the genealogy is (as I described it elsewhere) vertical, but then in v. 16 it becomes a horizontal genealogy, expanding outwards to describe the clans of Levi across the three main groupings: Gershon, Kohath and Merari. The vertical genealogy indicates the author's interest in the high priestly line, and the transition back to a horizontal genealogy permits the author to broaden the context in preparation for his next topical focus.
The reference to Amminadab is interesting because in most cases, the name Amminadab is usually affiliated with Judah (Num 1:7). In fact, this is the only place in the bible where Amminadab is called a son of Kohath. Usual listings (e.g. Ex 6:18) and even later in this same chapter place Izhar as a son of Kohath instead of Amminadab. It's possible that Amminadab is called a son of Kohath because Amminadab's daughter married Aaron (Ex 6:23), and more to the point, the Chronicler may wish to associate Judah and Levi together to justify David's interference with temple affairs as well as the temple's placement in Judah.
I also think it's noteworthy that Samuel is called a descendant of Kohath. This is actually contrary to 1 Samuel 1:1 that specifically calls Elkanah (the father of Samuel) an Ephraimite. It's interesting because I remember one of the most confusing parts of the book of Samuel is why Samuel, an Ephraimite, would be permitted to minister in the temple and to remain in the tabernacle before the mercy seat (1 Sam 3:3). It would actually explain a lot if Samuel were really a Levite. I'm just skeptical because this is the first time it's ever implied that Samuel was a Levite and the book of Samuel is much closer to a primary source than Chronicles. Chronicles shows that it's willing to bend the truth to make certain political or theological points, and it may be doing that here also. However, many sections of Chronicles shows that the author of Chronicles had access to a lot of what are now missing works, so I don't think it's impossible that the Chronicler is deriving this section of the genealogy from a historical source that no longer exists, which could also explain the deviation from 1 Samuel.
The Chronicler is especially focused on Samuel because verses 22-28 and 33-38 form a chiasm centered around Samuel and his son Joel. Verses 22-28 give a genealogical descent from Kohath to Samuel, and verses 33-38 give a reverse genealogy from Samuel back up to Kohath. Samuel is very important in the kingship narrative because he's the man who anoints both Saul and David. The Chronicler begins his historical narrative in chapter 10 with Saul and David, so it's appropriate that the genealogy would emphasize the men related to those stories. The genealogy in verse 33 also associates Heman (an important musician) with Samuel (an important priest and prophet). It's a subtle point, but I think it says a lot about the Chronicler's intentions with this book: he wants to tie together the legendary heroes of Israel's past with the "modern" (in his time) worship and temple ministry.
Verse 31 (in the middle of this chiasm) also transitions the genealogy to the next major section, the musicians. Besides Heman's association with Samuel, this section just as importantly declares that the musicians are appointed by David (v. 31). There are three chief musicians, one from each clan of Levi, and we are given their genealogies as well. Heman (from Kohath) is the chief musician, perhaps because Kohath is also the clan of Moses and Aaron and the high priests. This at least partially establishes the equality of the Levitical clans, since they each have a representative, though Heman from Kohath is given the preeminence.
Lastly, this chapter concludes by listing the Levitical towns (which they received in lieu of a tribal inheritance), which is largely copied from Joshua 21. I'm not sure why this is mentioned here, except perhaps to remind the readers that while the other tribes received land, the tribe of Levi received their temple ministry as their inheritance. This could again be a subtle way of implying that the Levites are the only proper men to offer sacrifices and serve in the temple. I also want to mention that there are four groups of towns, the first going to the sons of Aaron and then the three clans of Levi. Even though Kohath is not the oldest son of Levi, he effectively gets a double portion of the Levitical towns because of the prominence of Aaron and the priestly line. Other than that, I don't think there is anything notable or interesting in the list of towns.