Can I keep track of key events at my browser?

Don't worry, I don't want to hack anyone.
I have finished exams and the 2nd semester begins in more than 10 days, so I am playing around...

So, I know how to code an application to keep track of key events, but I do not know how I can make it run in parallel with my web browser (chrome or firefox). Actually, since this is my 1st attempt, which browser is more suitable for this "game" of mine?

As I have it in my mind:
I start my browser and the program I have made to keep track of key strokes.
I am surfing around, typing blah blah blah.
At the end of surfing, I would like to have somewhere stored all this information. (in a .txt probably?).
Then, I open the .txt and see the data..

Can I really do such thing?

EDIT: I could just print them in the console log instead of a txt. Of course, I am would be happy if I could get just the codes of the keys.

The question: Can I have the program running in another tab, but collecting key events that happen in other tabs?

Or maybe, I should just search for a keylogger? But that way it won't be fun..

You may want to look into how to make extensions for the browsers you wish to use. Perhaps there might be answers in such things.

Thanks for the answer :)