I'm planning on writing an Alternate History story revolving around it, and it would be eminently helpful to know how things happened in the original timeline when writing a new one. Also, is there anything you would suggest as reading for this?
If you know French, the historian Henri Guillemin has done an INCREDIBLE breakdown of it for Swiss television that you can find on youtube.
John Merriman from Yale teaches on this topic in one of his courses that you can find online. John is the most left and most outspokenly so that you'll find. He's also probably the best for a contemporary approach. If you do nothing else, watch his course.
If not, the Horne book that the Alaskan referenced is a little old and a little bourgeois, but it's the sine qua non of anglophone scholarship on the Commune. Robert Tombs "The War Against Paris" and "The Paris Commune 1871" is a little more sympathetic, and frankly less pop-history. For French, Jacques Rougerie is the most well-known, and has written 4 or 5 books on the Commune. You I have a literally 20-page long bibliography if you want me to send it to you. Most of the scholarship has been in French, and mostly from the 70s and 80s.
Robert Tombs said that you need to understand the siege of Paris of 1870-1871 in order to understand the Commune, so take that or leave it…
The Commune is an incredibly complex moment, and as Simone Weil said, "To learn the official history, is to listen to criminals"… Best of luck!
The classic book on this topic is The Fall of Paris, by Alistair Horne. I've a copy of the 1967 paperback edition on my shelf, and it's still extremely readable.