Alaska does have the western-most and northern-most points in the United States. Depending on how you think of it, the western-most point in the country could also be considered its most eastern point.
The reason that our 49th state has the country's western-most point that is still on the North American continent is that it includes the Aleutian Islands. The island in that chain that sits farthest west is Unyak, which is at longitude 179° 46'.
That longitude sits farther toward Asia than does any other point in the United States that is in North America. It is only 54' east of the International Date Line!
For that reason, some people consider Unyak the eastern-most point in the United States, too, since one could travel east across the continent from Alaska, then across the Atlantic Ocean, across Europe and Asia, and arrive there.
I don't think that's a very convincing argument myself, even though we can think of the International Date Line as the most eastern point in the world. It seems more of a reflection of common sense to consider the eastern-most point of the United States that is still in North America to be West Quoddy Head, Maine (longitude 66° 56' 59.2" W).
As for the country's northern-most point that is still in North America, you would go to Point Barrow, Alaska (latitude 71° 23' 20" N).
You can get to Point Barrow or West Quoddy Head by combining air or, possibly, train travel to the nearest major city (Portland in Maine, Fairbanks in Alaska) and then catching either a flight on a smaller aircraft or traveling by motor vehicle to those destinations.
Unyak is uninhabited. You could possibly try to secure a watercraft to take you there, but as far as I can determine there are no air or marine travel services to take you there.
Keep in mind, too, that this answer does not take into account U.S. territory that is not in North America.
