At least as early as 1553 when Sir Hugh Willoughby led an expedition to find the New World via the "Northern Sea route". His crew got as far north as the Kara Strait, north of Russia, encountered ice, and had to turn back.

Several other explorers tried to find the "Northern sea route" over the next fifty years, including Stephen Burrough, Willem Barentsz, and Henry Hudson. A common theme was that these explorers thought that if they went north in the summertime, it would be warm enough that the ice would melt, but they invariably encountered ice, and had to abandon their plans.

There were, of course, Viking voyages that entered the Arctic Circle earlier, with their discoveries of Iceland and Greenland before the year 1000. And there may have been some exploration going back to antiquity. But as far as actual exploration goes that produced surviving written accounts, they began in the mid-1500s.

It wouldn't be until after the Industrial Revolution when humans reached the North Pole.

Further reading:

The Great White North: The Story of Polar Exploration from the Earliest Times to the Discovery of the Pole by Helen Saunders Wright

Through Siberia, the Land of the Future by Fridtjof Nansen, translated by Arthur G. Chater (includes timeline)