Much is said about the Viking and Spanish discoveries of the Americas, but what of the Inuit? If I've understood correctly the peoples of Nunavut, Alaska, Siberia, and other Artic Ocean facing lands have a similar language (Inuit-Yupik) and, in my lay opinion, were able to the North pole between Canada and Siberia without much trouble.
Is there any evidence of this? When the Russians expanded into Siberia did they end up capturing populations of Native Americans? Inuits who had possibly crossed to Siberia, maybe Inuit merchant communities?
SO are you asking if the the Inuit "discovered" North America or if they were the first to reach the north pole?
If you are asking if the Inuit discovered America there are two possible ways to answer but the answer to both are not really and probably not. If you are asking if they discovered it outright the answer is not really because what we really mean when we say someone discovered something is "added it to the existing body of western academic knowledge" which they clearly did not.
If you are asking if they were the first people to reach north America the answer is still probably not. There were two or three major migratory movements to north America in the past 30,000 years or so and the proto-inuit probably came later. The first groups to cross the Bering land bridge were probably more closely related to the Indians of the rest of the Americas (if I remember correctly), the precursors to the Inut probably arrived from Siberia much later.
As for reaching the North Pole I doubt that as well. Why would they go there? They would either have to reach it in the winter and get back to mainland before the summer melt but why would they? They primary food source was in the water, you can't really hunt on a completely frozen ocean. They also could have gotten stuck in the summer when the ice would have been thinner but I doubt many would have intentionally done that and chances seem high that anyone who was separated from the larger population for so long would probably die.
I'm no expert in Native American history, I read Charles C. Mann's 1492 which does address this and took some courses which addressed migration to north america in college but hopefully this response answers part of your question.
Peopling of the Americas archaeologist over here, WhoH8in gives you a great answer. One disagreement- I don't think we can say whether or not the Inuit went to the North Pole. That would be a topic for more recent studies than what I'm interested in, so I don't know specifically. However, I'm a fan of the idea that absence of evidence is not evidence of absence.
Moving on, you asked about Russians and Siberia. There were (and are) indigenous Siberian populations (e.g. Tungusic, Samoyed) in the great snowy northeast. The Russians did encounter them, and did some not so nice things to them and their culture. My personal research interests are not focused on this subject, so that is about the extent of my knowledge about the Russian treatment of indigenous Siberians.
These populations appear to be genetically related to Native Americans, to varying degrees. It is commonly accepted that the indigenous Siberian populations are ancestral to the indigenous American populations. This means that way back in the day (thousands of years ago) some people from Siberia crossed the land bridge between North American and Siberia and then populated all of the Americas. The land bridge only existed during the last ice age, until ~10,000 years ago. After that, the area between Siberia and North America was the Bering Sea- crazy cold ocean water. Therefore, there probably wasn't a whole lot of contact between Siberia and North America after the last ice age ended. While the land bridge was there, it was highly likely that people were going back and forth between the two places. In fact, a linguistic study has recently proved that to be the case.
Does that answer some of what you were asking? Do you have additional questions? You can also head over to /r/AncientMigrations to read some articles about the peopling of the Americas.