Hey!
Do you know a bit about JS?
I’ll try to write down the general outline, so you can maybe figure it out yourself.
I’m not sure how this ties into chrome extensions, so I can’t help you with that, but hope this is helpfull.
The API only updates once every 30 seconds or so.
So you’d start with checking for updates every 30 seconds.
Javascript has a setInterval(x,y) function that repeats a given function x every y seconds.
Now, in this loop you probably want to load the channel data, like you did.
You load the data in a variable as a JSON. ( i assume you know this much, if this is too hard, maybe you can use Jquery for this?)
Okay great. We got the data!
Now all you have to do aswell, is check the JSON object called “stream”.
If this isn’t null, the channel is online!
so you can simply compare it to null 
(I think it’s the string “null”)
So now you figured out if the channel is on.
You probably want a variable that keeps check of the last read, so you don’t spam messages about the streamer being online or being offline. so if the new result is different from this global variable, you can show a message.