After the Commonwealth of Liberia declared independence in 1847, immigration to the new country was largely orchestrated through the American Colonization Society (ACS): 4,000 migrants between 1848 and 1854. The ACS raised funds from three different types of sources:
The leaders of the ACS petitioned the federal government for financial assistance to develop a steamship line after the Commonwealth declared independence in 1847, but the Congress only agreed to establish a US consul in Monrovia, the capital of Liberia.
By 1867, the ACS had collected more than $2.5m for the cause and helped roughly 12,000 people in total to immigrate to Liberia.
Source: Cassell, C. Abayomi. 1970. Liberia: History of the First Republic. Chapter 4: Abolition or Colonization?
Sadly I'm late to an appointment and don't have time to answer the second part of your question, which is incredibly complicated. I'll try and check back soon.
So far as I'm aware, the American Colonization Society received the great majority of its money from private donors. It lobbied the Monroe administration for cash set aside for the suppression of the slave trade off the African coast to be used for a settlement. Monroe's cabinet divided on whether that was a good idea or not, but he ended up going along. As the negotiations, which included a navy officer pulling a gun on a local ruler and declaring that the Americans were totally his buds, were conducted under government auspices rather than privately, and the United States maintained sort-of jurisdiction[1] over Liberia I don't think that the money ever went through ACS hands.
Public funding for colonization was extremely controversial, because there were serious fears that if the government could buy slaves then it could buy them on a big enough scale to cause a plunge in their price. That would induce enslavers to sell while they could rather than take a bath on all the capital they had invested in people. This, they imagined, would lead in short order to a cascading failure of slavery throughout the South which would probably overwhelm government coffers and thus leave the white South with the nightmare scenario: large numbers of free blacks living amongst them. That just had to lead to some kind of race war, given half a minute.
But this does point to the double-bind that hindered the ACS. If it's an antislavery organization, as some people very much intended it to be, then the goal is getting rid of slavery. (By getting rid of the slaves, mind.) then the obvious place to go looking for colonists is out on the plantations. Where those slaves are the enslaver's money. Very few of them, as aforementioned, wanted to sell.
So you go to the free blacks...who mostly don't want to leave. A great many of them were born here. They had friends and family. They might own property, rarely even in substantial amounts. Why should they leave? Yes, their situation was often very far from ideal but if home sucked then you could make it better. Liberia was the other end of the world. Many of them in the Northern states were involved in one way or another with helping fugitives and other antislavery efforts. If they go away, what happens to that program? Are their loved ones left behind stuck in slavery in perpetuity, never to be heard from again?
There's a constituency supporting the ACS, though by no means the whole of their support, for whom that's the actual plan. On paper, colonization will get rid of all the blacks. But really it's going to get rid of the free blacks who are a threat, sometimes an existential one, to their system of white supremacy. Everyone white was equal and free; that's what being white meant. Everyone black was inferior. Through their inferiority, they made whites equal. Through their being enslaved, they made whites free. Those attitudes aren't universal in the white south, or limited to it, but they're clearly the dominant discourse.
Thus the people that the ACS could, in theory, make go are largely unavailable. The people who are free to go aren't interested in large enough numbers to really make a dent in the population. And either way, it arouses sufficient enough outrage in the South (including to proposals for it by northern states that frame slavery as a shared national responsibility) that I don't know of any significant federal funds going to the effort direct, aside those that bought the land in Liberia, until the outbreak of the Civil War. Plans to direct revenues from selling western lands, the Jacksonian budget surplus, and the like were all torpedoed as crypto-abolitionism.[2]
There is some state action here, though it's not all connected directly to Liberia or the ACS. Virginia and Maryland both embarked on colonization projects. Maryland's 1832 plan rejected Liberia for its reputation as a death trap, they appropriated $200,000 for free transport, some kind of support on the far end, and advertising the opportunity to the state's free blacks. About a thousand people took the trip, which is barely a drop in the water. The independent Maryland colony foundered and was annexed to Liberia in the 1850s.
By contrast, Virginia had a more forceful plan and did less with it. Nat Turner's uprising convinced some Virginians that they ought to be clear of slavery. Starting immediately, William Henry Broadnax proposed, Virginia would deport 6,000 free blacks a year. Either the state or Washington would pay for it. Give it ten years and they would be out of free blacks. They wouldn't be asked, they would be sent. Then Virginia would start in on the slaves to the tune of 10,000 a year, for which they would pay the slaves' owners. It would be expensive and take eighty years, but Virginia would be all-white and, as a consequence, all right.
There were internal divisions that complicate all of this, many of them also having to do with Virginia's new constitution and malaportionment. The mostly-white far west of the state didn't much care to have free blacks gone, but would liek to be rid of slaves. The more enslaved areas in the Tidewater very much wanted quit of the free blacks and might get around to the slaves someday. Ultimately, in 1833, a proposal did get through the legislature for $18,000 a year for five years earmarked to colonize free blacks, but the law didn't permit the seizure or deporation of slaves, nor require it of free people. They, as they had else where, chose not to go. I'm not sure if any of Virginia's money was ever used.
The outbreak of the Civil War put most of the proslavery colonizations and anti-colonizationists out of the Congress, which meant things turned around quite a bit. In 1862, Congress appropriated $600,000 for the purpose. The measure was popular in the Northwest and Border States, where it was seen as a good way to show the public that the Republicans were solid white supremacists, not soft-hearts or racial egalitarians as the Democrats charged. The radicals were not happy, but most of them went along in the name of party solidarity.
This didn't mean that the colonists would go to Liberia, necessarily. The Lincoln administration had felt out Guatemala and Honduras, who were not interested and suggested instead that the American West be used. There was talk about placing a colony on the Amazon. A Danish diplomat tried to get Washington interested in St. Croix, which had a labor shortage thanks to Denmark's abolition. They considered Columbia, where the colony could include some strategically useful coal mines. Eventually there was a short-lived colony off the coast of Haiti. It turned out that the guy in charge treated it as a personal money-making scheme-cum-plantation where his colonists, who would basically be slaves, would grow him cotton. They ran him off the island and some went to Haiti proper on their own, but the War Department ended up picking up the survivors after less than a year.
[1]So far as I am aware, this mostly consisted of occasional visits by the Africa Squadron, which was entirely absent some years and made only token visits in others but did occasionally manage a good showing.
[2]Which is also what went on behind a lot of the general strict construction/state's rights rhetoric. If the government has the cash and power to manage great endeavors like national roads, or even banks, then it has the power to emancipate. This was an accepted orthodoxy among Southern radicals (albeit at the time more Virginians than Carolinians) back when Calhoun was an establishment guy promoting a big national road project, but of course South Carolina's most famous son came around. This all probably sounds a bit paranoid, and it was, but there were serious arguments that the simple act of debating slavery in the Congress would bring bloody slave revolts. The notion could extend even to private conversations and informed quite a bit of the effort to make sure slaves couldn't learn to read. Because slaves were always happy right up until someone told them to be unhappy, or so they thought.
Sources:
Don Fehrenbacher's The Slaveholding Republic and Robert Pierce Forbes' The Missouri Compromise and its Aftermath touch on the founding of the ACS and the story of the naval officer with the gun is from his footnotes. Fenrehbacher also has a very thorough discussion of the effectiveness, or lack thereof, of American efforts to combat the African slave trade.
William Freehling goes into the fears that colonization would lead to emancipation and the difficulties in finding colonists in the first volume of Road to Disunion. The securing of slavery through illiteracy is a theme he also develops in Prelude to the Civil War.
Eric Foner's The Fiery Trial discusses Lincoln's colonization misadventures.