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The best country to be able get the visa easier than the other countries to get to the north pole (Arctic area) is Russia. Now, how can I get there cheaply?
Often trips to North Pole start at Spitsbergen which is mostly Norway(one village is Russian), with one note - they are either helicopter or airplane trips and costs sthm like 15000 euro. If you would like to travel from Russia then icecutter is your choice and it is much more expensive(though longer) starting from 24000 USD and to infinity 8)). Here is link to reliable travel company RussiaDiscovery (translate it via Google translate). Trip starts in Murmansk. Also look at tema's post about North Pole - Tema North Pole(again Google translate)
Travel there is expensive. There is no commercial aviation or transport there. Just military or travel companies.
Not sure about the cheapness of this but here are the options:
According to Wikitravel getting there would run $18-$25k. The Arctic Expeditions doesn't provide the price but the only thing that they do state is that the Dog sled tour starts from Norway.