To see the northern lights, you only need some basics:
- It needs to be dark. Very dark. So, away from light pollution (any lights on the ground nearby or in the distance). Getting far away from any town is best. Long nights are best, and far north in the winter helps (= long nights). Be up late at night when it's darkest. Dress warmly, as you will probably be out in the cold for hours.
- There needs to be auroral activity - solar wind - which is impossible to predict (it can be forecast on very short notice, like within a couple days, when the sun puts out more juice than normal). This is pretty much a crap-shoot, unless you can drop your life and fly to somewhere in the far north on very short notice - so not practical for tourists.
- The sky needs to be clear. Any clouds mean no view of anything in the sky. Something to consider when visiting northern places in the winter - in winter time, it's often cloudy in these places.
- There is absolutely nothing special about Iceland, or anyplace else for that matter, that makes it "the place" to watch the aurora (other than major marketing efforts). Iceland is about as good (or as bad) as anyplace else on earth at roughly the same latitude - plenty of places in Alaska, northern Canada, most of Scandinavia are equivalent.
- You need to be either patient (maybe very patient - ie prepared to wait days or weeks), or you need to be very lucky. A short visit (to anyplace) with the expectation of seeing a good aurora display on your one or two nights there is a recipe for disappointment.
If the conditions are poor, you're not going to see anything special in the sky even if you go to the north pole.
If the conditions are great, you can see the aurora from your back yard.
I've spent many months (including winter months) in Alaska (including some pretty remote places). I did see some minor aurora activity there a few times (also spent many hours staring at the cold dark sky). It was interesting, but probably not terribly impressive to most people.
The most spectacular aurora I ever saw, with vivid, intense, wild shifting colors filling the sky for hours, was when I was in Detroit, Michigan. That aurora was a doozie - it was visible as far south as Mexico City. I would not suggest anyone travel to Detroit (or Mexico City) for a few days with hopes to see the sky show.
Mostly it requires being in the right place and then getting lucky.
I think there are a LOT of deceptive travel businesses running "aurora tours" that leave visitors disappointed. If you want to visit Iceland to see that country, great, go do it. But don't count on seeing the aurora unless you're prepared to be there a while. Considering how expensive Iceland is, if I primarily wanted to see a great aurora display, I'd go to someplace with less of a marketing machine in place - say, Whitehorse (Yukon) or Yellowknife (Northwest Territories) or Anchorage/Fairbanks/Nome (Alaska) - and plan to stay for a week or two in the mid winter. You will still need some luck even there under the best conditions. Bring warm clothes, patience and some reading material.
No disrespect intended towards Iceland or their very effective tourism industry, or anyone who has gotten lucky there.