I thought about this during my history class today. To me, it doesn't make much sense as it's far away from the other states and, of course, it's very cold up there.
The US tried to unite Alaska with the rest of the country by acquiring parts of Western Canada from Britain in 1869 as part of the Alabama Claims. The US made claims against Britain for damages caused by the British selling warships to the Confederates during the US Civil War. One of the warships was the CSS Alabamba, for which the claims are named. In the build up to the negotiations over settlement, many in Congress wanted to demand British Colombia instead of monetary payment. Britain didn't bite and ended up paying $15.5M to settle the issue.
Edit: changed 1969 to 1869
To add onto /u/The_Alaskan 's point about manifest destiny and territorial acquisition, the Pacific Northwest was one of the last multi-European territorial struggles in North America. Britain and Russia had already been at odds in Crimea, and would jockey for influence in the Great Game.
As /u/playingthelonggame pointed out, there had been US sentiments since at least 1846 that the British-held Columbia District should be added to the Union, as expressed by the slogan "54°40' Or Fight!", which advocated taking the Oregon Territory all the way from the California border to the southern tip of the Alaska Panhandle. Expansionists hoped that the Alaska purchase, by encircling British Columbia, would help the US acquire that land, too.
The Russians sold Alaska for 2¢/acre partly to quit the competition with the British for Pacific region influence, and simultaneously boost the Brits' rivals for "British Oregon", namely the USA.
Besides, better to sell Alaska for cash than lose it for nothing: I haven't a better source for this quotation than wikipedia, but the tsarist Grand Duke Konstantin wrote
"we must not deceive ourselves and must foresee that the United States, aiming constantly to round out their possessions and desiring to dominate undividedly the whole of North America will take the afore-mentioned colonies [Alaska] from us and we shall not be able to regain them."