1. Our family watched Hamilton a week ago (what better program to watch on the 4th of July?!), and still the songs are in my head. Stating the obvious, Hamilton is an amazing piece of musical theater, and thus far it is the best cultural thing my daughters have introduced me to (through their endless listening to (and singing of) the soundtrack, even though I was only up for listening a few dozen times).
2. Hamilton is musical theater, not history. As history it is flawed on many levels. I won't link to articles, as you should google "historical inaccuracies in Hamilton" or something along those lines and start your own investigation (it's good for you). In your investigation, you'll see errors of several types. First, there are the simple chronological inaccuracies -- for example, Philip's duel happened after the Election of 1800, not before. Second and more significantly, there are inaccuracies that advance the story but which are partially or completely fictitious. For example, Burr and Hamilton weren't friends and rivals for all those years -- there's no evidence they knew each other prior to the end of the war when they both were lawyers in New York. Also, the Election of 1800 was far more complex than portrayed by Hamilton (it started with Burr being Jefferson's running mate, not opponent), and while Hamilton helped broker Jefferson's victory, his wasn't the decisive vote (heck, because he wasn't in the House of Representatives, he didn't even have a vote) -- it was Rep. Bayard from Delaware (it's ok to say "who?"). A third type of inaccuracy involves the omission of facts to portray Hamilton in a better light. Perhaps the most noteworthy is the whole penniless immigrant characterization that occurs throughout the play (aided by his being portrayed by a POC), which, together with the noting of his involvement with an anti-slavery organization and his friendship with John Laurens, obscures Hamilton's more complicated relationship with slavery. [Ok, I lied -- one link]. Fourth, there's the matter of how the portrayal of Hamilton is affected by Miranda's choice of source material -- Ron Chernow's book is quite popular, but it's also controversial among historians.
3. If your kids are old enough to watch Hamilton, they're old enough to learn about the play's historical shortcomings. Share what you learn about the actual history with them (or look them up together) -- critical thinking is one of the most valuable gifts you can give your children, and for me it's the best way I know to repay my daughters for introducing me to Hamilton in the first place.
Bonus thought -- Today is the 216th anniversary of the Hamilton-Burr duel. Hopefully your research will lead you to understand why the Election of 1800 wasn't the principal reason behind a duel that occurred nearly four years later.