I bought a Yamaha FZ6, and I was wondering about counter-steering. How much do I need to think about counter-steering?

depends how well you can ride.
If you have the basics then you can consider counter steering.
this can be helpful in everyday riding but essential when giving it some.
if you have been riding long enough you will find that you are doing it anyway

Better to think about counter-steering all the time. Go through your cornering technique in your head, from start to finish and emphasize the importance of both 'assertive' and 'passive' counter-steering. Simply leaning into a corner with your upper-body and pushing down on the inside bar (passive) is sometimes not enough, often the need to pull hard on the outside bar (positive) is obvious.

Don't rely on the explanation of 'look where you want to go' or 'just lean the bike', because while these rules are fine for a relaxed rider at moderate speeds on open roads, it doesn't quite cut the mustard when you're faced with oncoming traffic straddling the center line as you turn into a fast left-hander at 120km/h. Unless you think about counter-steering, understand it totally and rely on it in all weather conditions you may instinctively do the wrong thing when faced with a hazardous situation, and that split second decision will either save you or kill you.

Most people counter-steer automatically. You don't need to worry about it. Concentrate on your preparation for the corner and shifting your bodyweight, get your body position right and the counter-steering will happen automatically. Give yourself time to move about before you turn, and don't make any violent movements when you're hard on the brakes.