Can I upload NBA live game's screen recordings on YouTube?

Technically it is possible for sure, but I am not certain that it won't be deleted in one day. Most probably many rights will be violated and will not be able to get monetization from it. Anyway if you are still interested in this you can record your screen very easily using a special app. Here is the link https://screencapture.com/. I like this app because it has an online version, so you just record what you need and save the video on your computer, no need to download the program, but they a desktop version as well.