You know, I'm a historical sociologist (though M.A. in History also), and I've read most of Chernow's biography of Grant. I really enjoyed it, and came away with a very strong sense of who Grant was. However, I read it (and yes, I should finish it -- I got up to about 1875) just after reading Fallen Founder by Nancy Isenberg, a biography of Aaron Burr. I'd say the difference is that Chernow is a biographer, but Isenberg is a historian. Chernow does a great job of giving Grant historical context in terms of events, but Isenberg sets Burr (no relation to me, and I've traced our family lines back to the 1630s) within the deeper historical trends of the time. This may be because she had so little to work with, since most of Burr's personal papers went down in a shipwreck in 1811, along with his only child, Theodosia Burr Allston (on whom he doted, so this was a tragedy), but I still think Isenberg would have contextualized him just as deeply even with all of his papers. (Yes, his daughter was his only legitimate child -- Burr was like Franklin with the ladies, and there are many candidates for Burr's illegitimate children).

All of this is to say that I think Chernow's Hamilton biography will be very much like his on Grant: very tightly focused on Hamilton, what he did, how he responded to events, etc., but not drawing a wider panorama of the times in which Hamilton operated in order to help you understand him better. On the other hand, Isenberg's book on Burr was pretty short, however deeply sourced and historiographically grounded it was (and it was), while both of Chernow's biographies (on Hamilton and Grant) are huge, even massive, books, with tons of detail. You will walk away, I think, with a strong sense of who Hamilton was as a person, but not so much who he was in his times.

Sorry I'm not answering your question directly, but it might be a useful answer anyway.