I responded to a similar question a few months ago. I've copy-pasted that reply below, and I'll expound a bit in a subsequent comment.

There was a push, but it didn't happen because of Canadian Confederation, the settlement of the Alabama claims and divided American politics.

One of the most frequently cited papers on this topic is David Shi's 1978 work, "Seward's Attempt to Annex British Columbia, 1865-1869" which was published in Pacific Historical Review. You can also refer to Richard Neunherz's "Hemmed In: Reactions in British Columbia to the Purchase of Russian America," which appeared in the July 1989 issue of Pacific Northwest Quarterly and Donald Warner's 1960 book, The Idea of Continental Union: Agitation for the Annexation of Canada to the United States, 1849-1893.

The key things to understand are that at the time of the Alaska Purchase, there was a widespread belief that the United States would attempt to acquire "British Oregon" through purchase or force. In 1860, Eduard de Stoeckl, Russian minister to Washington, D.C., wrote that if Russia sold Alaska to the United States, "British Oregon would be crowded on the northern side and on the southern side by the Americans and would escape with difficulty from their aggressions."

America was very much expansionist in the years after the American Civil War. William Henry Seward, the Secretary of State, was a proponent of the Manifest Destiny ideal, but he supported the idea of purchase rather than conquest. You know about the Alaska Purchase, but Seward also explored the idea of buying Greenland, Iceland, and various Caribbean islands (most notably the Danish Virgin Islands).

At the time, however, the U.S. government was riven by a gaping divide between the two wings of the Republican Party. The Radical Republican wing, largely a product of the 1864 U.S. election, took a hard line on postwar reconstruction, and it viewed President Andrew Johnson's approach as much too moderate. This divide was so great that it led to Johnson's impeachment in 1867.

This was happening at the same time that Seward was attempting to push his imperialist ambitions, and the Radical Republican distaste for Johnson extended to the rest of his administration. Thus, Seward's attempts were foiled by internal dissent. Only the purchase of Alaska, aided by Russia's eagerness to sell and the application of liberal bribery, passed the U.S. Congress.

At the time of the Alaska Purchase, most Americans were in favor of Seward's ambitions. A survey of American newspapers at the time of the Purchase reveals that 45 percent of editorials on the Purchase also included the belief that the United States should acquire British Columbia as a bridge to its new possession. All but a handful of the editorials (and those in the most rabid Radical Republican newspapers) viewed the Alaska Purchase positively.

Debates in the U.S. House of Representatives, the topic of annexing British Columbia repeatedly came up during talks about the Alaska Purchase. 13 of the 31 addresses on the Purchase referenced its effects British Columbia, and several speakers said the Purchase would speed annexation of British Columbia.

In the immediate postwar years, the United States was also engaged in negotiations with Great Britain over the Alabama claims. During the American Civil War, Great Britain had sold or manufactured a series of commerce raiders that the Confederate Navy used with great success, destroying American shipping around the world ─ from the waters off the U.K. to the South Pacific and even the Arctic Ocean.

In early 1867, there were indications that Britain was prepared to pay the United States to settle claims that it bore responsibility for the destruction of these American ships. Seward pressed for more, however, claiming in his letters that Britain's recognition of the Confederacy as a belligerent extended the war. That would have opened the door to massive indirect claims.

In 1866, the New York Times quoted a government report that proffered the idea that Britain ─ instead of paying the United States to settle the Alabama claims ─ might instead give the United States territory worth the amount of the claims. "If Great Britain desires to propitiate this country," the report stated, "would it not be her true policy to cede to us a portion of her remote territories, valuable to us, but of little value to her? Were she to cede us Vancouver's Island and British Columbia ... might she not easily bring our claims to a peaceful solution?"

In the colony itself, people were divided. British Columbia was in a recession at the time of the Alaska Purchase ─ its economy was down, people were emigrating, and the government was an undemocratic farce. In an era when border controls were almost nonexistent, plenty of Americans lived in British Columbia, and there was also a sharp divided between the Vancouver Island residents and the mainlanders. Until just a few years before, they had been separate colonies.

Of the six major newspapers in British Columbia, three endorsed annexation by the United States. All were on Vancouver Island. Mainland publications largely rejected the idea. The Victoria News stated that a transfer to the United States "would be received by a majority ... of every nationality, with ... unmingled satisfaction ... if not loud demonstrations of joy."

The Times-Colonist reported that talk of annexation was "heard at every street corner," but the newspaper itself said it remained in favor of Canadian confederation rather than annexation.

A petition began circulating in favor of American annexation, but it turned into a fizzle as the summer of 1867 continued. Opponents of annexation argued successfully (if not truthfully) that the annexation movement was being funded by "outside money" and didn't truly represent the colony's inhabitants.

Nevertheless, U.S. Army officers visited British Columbia in the fall and turned in a report that said two-thirds of the colony was in favor of annexation. The report contained so much detail that it could only have been constructed in close consultation with higher-ups in the colony, even if it didn't mention those higher-ups by name. The officers found annexation talk everywhere: "It did not become necessary in a single instance to broach the subject of the cession of that territory to the United States, for it was the constant theme of conversation ─ obtruded upon us everywhere, and upon nearly all occasions when we came in contact with the people, either officials or citizens."

In the end, however, all this furor and planning amounted to nothing. Canadian confederation in the same year as the Alaska Purchase generated enough local opposition to annexation that it never happened locally. The impeachment of Andrew Johnson hamstringed Seward's expansion efforts, and the election of Ulysses S. Grant replaced Seward in 1868 with Hamilton Fish. Three years after he took office, Fish settled the Alabama claims with an international arbitration tribunal that awarded the United States $15.5 million.

After the settlement of the Alaska Boundary Dispute secured the Alaska-Canada border, there was much less of a notion to expand the borders of the United States at the expense of Canada. In place of that idea were a series of plans to establish overland transportation links between the 48 states and Alaska. During the Klondike Gold Rush, there were dozens, if not hundreds, of vessels wrecked while traveling the Inside Passage. Those who attempted to travel overland had to endure months of incomprehensibly difficult travel and impossible conditions.

Between 1897 and 1935, there were many railway and roadway plans floated by northern promoters. These never went anywhere ─ reluctant bankers and reluctant politicians doubted their economic and engineering feasibility, and so none were ever built. In the 1930s, attitudes began to change. In the United States, infrastructure construction was on the upswing, courtesy of a surge in federal spending. The growing threat of Japan, and continued high gold prices encouraged a new surge of serious interest. Plans were drawn up, but the Canadian and American provincial and federal governments were reluctant to pay for the project.

That changed with the outbreak of World War II. With Japan on the advance in 1941 and early 1942, it seemed possible that Japan's naval superiority could result in Alaska's coastal supply routes being cut off. An overland route, it was thought, would be less vulnerable to attack and thus important to the war effort. In hindsight, this was wrong, but with political pressure from military authorities added to that of Northern politicians, the road was funded and built, serving as a physical link between Alaska and the Lower 48.