Can I install my iPhone battery myself?

The short answer is: "no".

The long answer is: Sure, provided you understand that doing so will result in rendering your phone untouchable by Apple and that you understand it is impossible for you to purchase a genuine, new battery for your phone as Apple does not sell parts to unauthorized service providers. The best you'll be able to get is a knockoff, counterfeit, salvaged, or stolen battery.

Attempting this may result in losing functionality for Face ID, even if you do the job "correctly".

Just have it replaced by Apple. It's not that expensive.

This forum is full of posts by well-intentioned folks who have messed-up their pricey Apple devices attempting to save a few dollars.

As replacement batteries aren’t that expensive; it might seem to be an unnecessary risk to “learn how” on your own “only” device.

iPhone Battery Repair & Replacement - Apple Support