Russia colonized Alaska in the first place mainly because it was the only place they could reach that didn't have six other nations elbowing each other in the face to claim land there. We all laughed at Tina Fey, but Russia is pretty much right next to Alaska, a lot closer than any of the European colonial nations to the Americas, which just about makes up for how long and cold the winter is. There used to be a whole lot of seals and bears in Alaska, neatly making up for thousands of years of killing everything that could make a pair of mittens in Siberia. So people started making their way east, and making a fair amount of dough skinning the wildlife.
The Russian Empire got antsy about Alaska not too long after the Crimean War against Great Britain, from 1853 to 1856; British Columbia was right next door, and when they found gold people started coming there enough that the crown established a full-fledged colony. Tsar Alexander II figured that next time Russia went to war with England, they'd probably lose Alaska anyway, so selling it would at least give them some cash for their trouble.
So they started trying to get some interest from England and the US in 1859, but we were busy with the Civil War (it hadn't started yet, but flames were in the air and it would have taken a miracle to stop) and England didn't much care, so nothing happened for a few years. In that time, Russia also borrowed a crapload of money, and now really needed to sell Alaska.
William Seward, Lincoln's Secretary of State, got a lot more receptive after the war ended, and agreed on a price of $7.2 million, a bit more than 2 cents per acre for a place twice as big as Texas. There was a lot of grumbling in the press over this, since we were hardly pressed for space or flush with cash as it was, but on the upside, Russia liked us a lot more, and hemming in British Columbia made annexing it eventually more of a possibility. (Spoiler warning: we didn't.)
Still, those two justifications probably wouldn't have been enough without the fur trade: Russia had set up 23 trading posts, and around 10,000 people lived in Alaska where the harvesting company was working. We started charging an annual $50,000 fee for being allowed to club seals, and $2.625 per skin; with a hundred thousand skins per year, it would be around twenty-five years to recoup the costs of the purchase, assuming more people didn't move there to hunt seals, and nobody found gold.
Of course, we did find gold in Alaska, first in Klondike and then in Nome. Around a hundred thousand prospectors trekked up there from 1896 to 1899, but less than half of them made it, since traveling up through Canada was an incredible pain in the neck at the time. Maybe four thousand people actually made any appreciable money on the gold rush, and everyone else ended up with plenty of pain, but no gain. Of course, the real victims here were the natives, since they weren't allowed to buy land (and thus couldn't mine without risking getting pushed off), and the miners wrecked the environment something awful.