I need to figure out how to do splits on my NES so that I can make timing my run less tedious. So my question is, what should I do?

Splits are generated with a timer program. The two most common are Livesplit and WSplit. Those programs run on your computer and allow you to make splits to track your progress. You can make as many or as few splits as you like for your game.

After that, you can use a capture card to capture the gameplay from the NES. You can buy a splitter if you want to split the signal to the cap card and to your TV (otherwise you'll be playing off a computer monitor). Then you just have to add both the capture feed and the split program to broadcasting software like OBS or XSplit. The broadcasting software can either allow you to stream to various services like twitch or to local record to a video file saved on your computer.

After that you simply start your splits when you start your game and split at the appropriate times. If I remember correctly WSplit and Livesplit default either the "Enter" key or the space bar to be the split key, but you can change those in the settings at any time. I suggest messing around a lot to find a split setup and layout that you like.

Hope that helps.