There aren't any major cities. There is only one city in Iceland, which is Reykjavík, and even that is more like a large town. Akureyri is the 'capital of the north' but it's not a city.
You could take a domestic flight from Reykjavík or drive yourself up there. You can make the drive from the capital to Akureyri in one day, about 5 hours. Even if you fly up, I would self-drive as it gives you so much more freedom, but get a decent car if you plan to come in the winter.
If you want snow, there is a lot more snow after Christmas than before. The daylight is OK by March, so I would say February/March if snow and ice is a priority for you. But if you've already seen the winter, why not try the summer? The summer is really magical in my opinion.
Must sees in the north are Ásbyrgi, Mývatn, and the swimming pool at Hofsós. Siglufjörður is also a nice little town and the whale watching is good in the north (season dependent).
If you can combine North Iceland with West Fjords, I'd definitely recommend it. The West Fjords are amazing.