Buy a ticket to Grise Fjord. That's pretty arcticy. (No plants bigger than mosses, lots of rock.)
Ok more seriously, if you want to go on your own, you need to acquire some skills first.
A: Get good at backpacking.
B: Now start camping above timberline.
C: Now start camping above timberline in winter.
A: Learn to cross country ski.
B: Learn backcountry skiing.
C: Learn about avalanches.
D: Learn to do extended trip at/above tree line in winter.
Strictly speaking steep slope skiing and avalanche safety aren't necessary, but high elevation travel is the most practical way to practice winter travel techniques if you can't afford frequent travel to northern timberline.
Much of the arctic is soggy wet in summer. Summer travel is very slow cross country due to swamp, muskeg, rocks, rivers and lakes. It's also unpleasant due to bugs. To a slightly lesser degree this is true of the boreal forest too. There is good reason that essentially all northern settlements are on rivers or lakes.
A: Get a canoe.
B: Learn to paddle it in rough water.
C: Learn how much gear it can handle. Two months food = 60 days food = 120 pounds per person if carrying fully dried food. This has no allowance for ruined meals, the odd tin of jam or sardines, food ruined by water, rodents, and bears.
D: Get good at portaging.
First & Second Aid
You are your own doctor. You need to deal with everything from upset stomach, mosquito bite overdose, minor burns, scrapes to dealing with serious cuts, sprains, and potentially broken bones. Second aid is the skill to keep the victim alive, and comfortable until help arrives, or recovery has happened.
The arctic has polar bears as well as grizzly bears. Both consider you as lunch, particularly in areas that are not hunted by the locals. Hunting also allows you to augment your food supply. See vilhjalmur Stefansson
A: Learn to shoot.
B: Get your hunter-safety course.
C: Go hunting each fall.
D: Learn how to shoot bears.
A few hundred days and nights of outdoor travel split fairly between these topics will put you in the position of not being a serious danger to yourself and others.
In general you will find that high elevations are very similar to arctic conditions, and a lot easier to get to.
Meanwhile, read up on the history and geography of high latitude lands.