Here is a brief answer from /u/bolanrox regarding how the actual money was moved for the Louisiana Purchase. Crazy enough, the transaction was handled by a British bank while Britain was at war with France.

Hey, I'm still alive! Just buried in enough work that when I came up for air it'd been long enough so I was outright logged out from Reddit (along with just a few inbox messages waiting for me, heh.)

Here's the short answer: originally, it was supposed to be 7.2 million in gold bullion out of Washington, but after a nearly 18 month delay it ended up going as a Treasury warrant to a DC bank to Barings Bank in London as a fiduciary. The saga of why this happened, though, is a much longer and more interesting story.

The United States had a less tempestuous relationship with Imperial Russia in the early to mid 19th century than with France or Great Britain, starting with John Quincy Adams' appointment and continuing to when relations became warm during the Civil War. Both Britain and France showed significant Southern sympathies until late in the war, but Russia had thrown in with the North and a unified United States early on. There were all sorts of reasons for this that weighted heavily towards European politics, most notably that the enemy of Great Britain was often Russia's friend.

In one of the more overt displays of mutual interest, Russia sent a good slug of her fleet to both Atlantic and Pacific American ports in early 1863 with the cover story of a goodwill tour - and Russian sailors very definitely enjoyed themselves - and the covert one being planned merchant raiding of British and perhaps French shipping if Russia went to war with either or both powers over Poland, with the possibility that if either power intervened in the American Civil War (Great Britain with the Trent in 1861 for instance) that the fleet might help out a bit there too.

So this was the relatively friendly diplomatic basis for where the United States and Russia were shortly after the war. /u/The_Alaskan has previously posted a terrific answer about why the sale took place from the Russian perspective; another major factor was that Russia had watched from afar as American settlers had more or less ignored territorial claims first in the Southwest prior to the Mexican-American War and slightly later on the West Coast when Great Britain and America spat at each other for a number of years over the Oregon territory and its boundaries. This was starting to happen to Alaska and was considered likely over the next decade or two; Russia in short had all sorts of reasons to get out and in December 1866 pretty much most the Russian powers that be had signed off on what had been gelling since the 1850s.

This timing worked out rather conveniently with the ascension of William Seward, who had been pushing for years for a variety of territory outside the continental United States. One bill that he sponsored while a Senator - designating uninhabited islands mined for guano as American territory - slipped through completely unnoticed but served as the basis for successful American claims on a number of Pacific islands that proved highly consequential in World War II.

Seward had long targeted Alaska as an ideal territorial gain for a variety of reasons, and come February 1867, probably thanks to Thurlow Weed - Seward's long time primary political patron - providing company to Russia's ambassador Eduard de Stoeckl rehabbing an injury in New York for a couple weeks after a voyage back from St. Petersburg, the long simmering deal was executed in a hurry. Over a two week period starting on March 11th, the two went from theoretical discussions to coming up with a price (5 million immediately raised to 5.5, with Stoeckl thinking he might get 6 or 6.5 million, which then got raised to 7 million more or less spontaneously by Seward to move negotiations along as they wrote the treaty) to cabling Russia for approval on March 25th, to Stoeckel visiting Seward at home on the evening of Sunday, March 29th to let Seward know that the Tsar had indeed done so - to Seward shocking Stoeckel by opening up the State department that night to get the treaty concluded at 4 am.

Seward had informed the Cabinet and gotten Andrew Johnson's tacit approval for $7 million - but only on the late evening of the 29th did he so much as bother sending his son (who was Assistant Secretary of State) to inform Charles Sumner, the chairman of the Senate Foreign Relations committee, that not only had negotiations begun but a deal had been reached! While Sumner ultimately supported the purchase, in the rapidly deteriorating relationship between Congress and the President in early 1867, this slight did not go unnoticed by him and others and played a role in the nasty fight over the payment in 1868. One of the minor sticking points in the wee hours of Monday morning was that Stoeckel had asked for payment to be made out of London - we don't know why, but bullion did need to be shipped - and in return for waiving most of them (including the relatively profitable franchise selling Russian Alaskan ice to thirsty San Francisco), Seward added another $200,000 to the offer for a grand total of $7.2 million.

I'm skipping most of the details over the ratification process - /u/The_Alaskan covers the most important point, which is that the purchase was widely supported and "Seward's Folly" came into the lexicon later when he tried to buy the Dutch Antilles - but worth noting was that 19th Century Congresses generally did not meet until a full year after their election, so it took a special session of the Senate in April 1867 (very unusually, they'd already convened a couple weeks earlier to squabble with Johnson over Reconstruction) to ratify the treaty, which wasn't particularly well received given who negotiated it and how he did so - Seward already was becoming anathema to most Republicans - but was generally agreed as a good deal by and ultimately passed 37-2 on April 9th. Stoeckel got paid only 25,000 silver rubles by the Tsar for his work, which initially annoyed him given he'd gotten such a better deal than anyone expected, and then it got even worse when he discovered he was on the hook for a nearly $10,000 telegram that he thought Seward had paid for!

But the House - which under the Constitution must initiate all appropriations - wasn't back in session until December to do the financing of the deal, and in the meantime relations between the Johnson administration and Congress had moved from hostile to the most toxic in American history. Meanwhile, American settlers (and soldiers) moved in over the course of that year...and on top of it, a nearly $400,000 claim against the payment was made by the widow of a merchant mariner, Benjamin Perkins, who'd theoretically acted as a Russian agent in 1855 by advancing the purchase of gunpowder and other supplies to them. The Russian government had stiffed Perkins on any payment for years, but his widow had gotten powerful backers in the House and Senate, probably thanks to promises that they'd get a percentage of what she got back, much like attorneys litigating on contingency do today.

Meanwhile, the political context had reached its nadir as the House impeached Johnson in late February 1868 (after multiple attempts by Thaddeus Stevens prior to this) and pretty much everything else on the Hill ground to a halt for months. This presented a dilemma for Stoeckel, who actually wrote back to Russia after an inquiry that they could either wait (his preference) or just outright offer Alaska for free! In May, the House Foreign Relations committee - with a number of supporters of the Perkins claims who noisily dissented - recommended approval of a payment, but that debate got stalled until July and the final 113-43 vote didn't take place until July 28th, with the treasury warrant that you've posted being presented to Stoeckel on August 1st.

Stoeckel in turn endorsed the warrant over to George Riggs (of the Riggs Bank, which lasted for almost two centuries as D.C.'s major financial institution), who then disbursed it to the London office of Barings Brothers, which handled Russia's foreign accounts. That was how the payment was processed.

However, it wasn't over; there was all sorts of perceived scandal and investigation afterwards about the lobbying that went on behind the scenes, with opponents at the time claiming that Russia only received about $5 million after all the various bribes had been paid out. It wasn't quite that bad, but George Riggs admitted during the investigation that he left $165,000 in the United States on Stoeckel's orders, of which about $135,000 can't really be accounted for but at least some of which is believed to have gone quietly to influential members of Congress.

Precisely who benefited from this graft is a mystery lost to time, but here's a fun bit of speculation on my part: David O. Stewart in Impeached presents a very good case that substantial bribes were paid to various Senators to encourage them to acquit Andrew Johnson. As William Seward was at the heart of the dirty money fundraising efforts for the trial, one rather nutty possibility is that Stoeckel reimbursed himself after being shaken down for that too! A couple decades back, someone even found a note from Tsar Alexander confirming that he'd OK'd a diversion of part of the payment to Stoeckel for undisclosed expenses, so the evidence is conclusive and it's probably an underlying reason why the payment method switched from shipped bullion to domestic Treasury warrant.

On the bright side for Stoeckel's ethics and pocketbook, the one disposition that we do know about is a $10,000 portion of it: a particularly litigious telegram company finally got their bill settled!

Sources: Seward: Lincoln's Indispensable Man (Stahr, 2012), Seward's Folly: A New Look at the Alaska Purchase (Farrow, 2016)

I've a follow up question; in British Columbia many of the indigenous tribes never ceded their land; what was the status of the Indigenous people in Alaska and did they have any say in the transaction (I'm assuming not) or is there any record of their protest to the sale? Were they even aware it happened?