I've written extensively on this topic, but the answer to your question is somewhat buried in a child of that previous comment.

The United States bought Alaska as an outgrowth of Manifest Destiny ideals envisioned by American Secretary of State William Seward.

At the time of the Alaska Purchase, it was widely (if not quite universally) believed that the United States would continue its expansion until it encompassed all of North America. This was not a far-fetched idea if you consider the course of American history to that point. The United States had grown through conquest and acquisition, expanding in 60 years from a handful of isolated territories into a nation stretching from Atlantic to Pacific.

Seward was a firm believer in the continental ideal, and as a rising Republican leader (he was the favorite for the Republican presidential nomination in 1860 but lost to Lincoln) believed that Manifest Destiny would be helped if the United States had a strong perimeter based upon four key points. This perimeter would secure American borders and allow it to project power east and west across the oceans if it wanted. This power might be economic, or it might be military.

These four points were Hawaii, Alaska, Greenland/Iceland and the Danish West Indies.

Seward believed all four points could be acquired without great difficulty or price, and he almost pulled the entire thing off.

Alaska was the first step, largely because the preliminary work had already been done. Alaska had extraordinarily close economic ties to California (San Francisco in particular), which supplied most of its food and in turn received ice from the Kodiak archipelago. (Ice might be an afterthought today, but in an era before mechanical refrigeration, ice was Alaska's most profitable export in the 1850s and 1860s, leading even the fur trade.)

During the Crimean War, Russian-American Company managers in Alaska had planned a fake sale of their assets and the territory itself to San Francisco businessmen. Such a sale would have kept the territory out of the hands of the British or French, who were launching an ill-fated assault on Russia's Pacific Coast. As it turns out, the deal was aborted when the Hudson's Bay Company and RAC officials in Europe managed to negotiate Alaska's neutrality. (The HBC didn't want to endanger its profitable lease of the mainland of Alaska's panhandle.)

Though the sale was never completed, the idea of a purchase remained in the minds of Americans through the American Civil War. Russia remained an interested seller, as the Crimean War had exposed the vulnerability of its Pacific Coast, and its defeat in the war meant it incurred significant debt. Selling Alaska would provide hard currency.

With an interested buyer and an interested seller, Seward was able to negotiate a deal.

The problem was Congress.

At the time of the Alaska Purchase, Congress was in a deep, deep dispute with the Johnson Administration. This dispute would eventually culminate in an impeachment vote (though the Senate would fail, by one vote, to convict). It took a significant amount of bribery by Russian officials to convince the House of Representatives to allocate the payment for the treaty of cession one year after the treaty was ratified in 1867.

That was Seward's only success, however. As relations with Congress continued to deteriorate, he lacked the political capital to get any kind of movement on his other goals. The acquisition of the Danish West Indies was foiled when the 1867 San Narcisco hurricane struck the islands as Seward was trying to gain support.

Seward's opponents in Congress poisoned the well after the Alaska Purchase was secured, and though he managed to compile a significant report in favor of the purchase of Iceland and Greenland, he could not garner legislative support before the election of President Grant in November 1868. The same held true of Hawaii.

The Secretary of State at the time, William Seward, is considered responsible for the purchase. The press at the time was not very favorable of the deal and it was called "Seward's Folly" at least until gold was discovered years later.

Here is the treaty between the USA and the Russian Empire:

https://memory.loc.gov/cgi-bin/ampage?collId=llsl&fileName=015/llsl015.db&recNum=572

I'm curious if anyone can shed light on why it was written in both English and French. It seems like it should've been written in English and Russian?

For some contemporary views on the subject, I found this political cartoon from Harper's magazine, April 20 1867:

http://www.harpweek.com/09Cartoon/BrowseByDateCartoon.asp?Month=April&Date=20

The map is labeled "Map of the Russian fairy land. Only $7,000,000 in gold" The caption also suggests that the purchase was a bit of distraction to the president who took over after Lincoln's assassination and presided over the first years of Reconstruction.

This New York Tribune article from April 9th, 1867 is equally disparaging:

http://chroniclingamerica.loc.gov/lccn/sn83030214/1867-04-09/ed-1/seq-1/#words=TREATY+Seward+treaty&date1=1867&sort=date&sort=date&date2=1867&searchType=advanced&proxdistance=5&state=&rows=20&ortext=&proxtext=Treaty&phrasetext=Seward&andtext=&dateFilterType=yearRange&page=2&page=3&index=13

This article suggests the reason for buying Alaska was for self glory and even suggests the Czar would have given it to the US for free.

Now, why did Seward try to buy Alaska? After the civil war, there was no longer any question of whether new US territory would become a slave state or not. This was a major reason why Cuba wasn't annexed as it would have become a free state and tipped the slave/free state balance. So at this point, Seward was looking to expand the US interest while at the same time Russia was looking to at least get something for the Alaskan territory it owned de jure before it was colonized and became American de facto.