How do I dynamically edit the html and css of a website that is NOT mine using javascript?

I'm trying to make a script that highlights certain divs/creates new divs on chess.com, a widely used website; this will be used to highlight moves that are suggested by an engine I will be making. I am really wondering how I'm supposed to get the website to read a .js file from my personal computer. I am aware this is quite an open ended question, so I am expecting open ended answers.