Hi Marten,
I found what looks like a fairly decent video on battery replacement on your phone. It goes step by step and gives you appropriate warnings when needed and also lists the tools used to do the repair. A T5 Torx driver looks like the most unique item you'll need.
[link|https://www.youtube.com/watch?v=qSWMFBcWYJM|HTC One A9 Battery Replacement Guide - RepairsUniverse - YouTube|new_window=true]
Mostly I'd say work slowly and carefully while removing the screen. If you don't have a hot air gun, generally a hair dryer will work fine, or you can splurge a few bucks and buy iFixit's [product|IF145-198-4|iOpener|new_window=true] that works quite well.
In addition, you'll need some adhesive to resecure the screen in place when you're done; precut is definitely preferred. I always pick up at least a couple when I buy them; sometimes it's easy to get them mispositioned and when that happens you don't want to have to wait until you reorder it. There are also times when you might forget something and have to open the phone back up again; I've done that a few times myself.
Here's an example of the adhesive; if you look around at places like Amazon, AliExpress and eBay you'll find plenty more.
[link|https://www.amazon.com/BisLinks%C2%AE-Brand-Housing-Adhesive-Replacement/dp/B01I3SNDJK|Amazon.com: BisLinks 2 x Front Housing LCD Adhesive Glue Tape Replacement for HTC One A9 : Cell Phones & Accessories|new_window=true]