It was divisive in Britain, not least because it had a direct economic impact on certain parts of the country — most particularly Lancashire and the industrial northwest, where the textile industry was crucial and where the decline in the supply of cotton from the southern US created an economic crisis. There was even a school of thought (albeit limited in influence) which advocated British intervention on the side of the Confederacy, in order to reopen the South's ports and the flow of raw cotton.
From James McPherson's peerless Battle Cry of Freedom:
In July 1862 the supply of raw cotton in Britain stood at one-third the normal level. Three-quarters of the cotton-mill workers were unemployed or on short time. Charity and the dole could not ward off restiveness in Lancashire working-class districts. [p. 548]
But on the other hand, you've also got a political perspective on the war that makes British radicals/progressives ardently pro-Union. Again, back to McPherson:
In this view, the issues of the American Civil War mirrored the issues of class conflict in Britain. The Union stood for popular government, equal rights, and the dignity of labor; the Confederacy stood for aristocracy, privilege, and slavery. [...] British radicals expounded numerous variations on [Lincoln's] theme. For a generation they had fought for the democratization of British politics and improved conditions for the working class. For them, America was a "beacon of freedom" lighting the path to reform. [p. 549]
Source: James M. McPherson, Battle Cry of Freedom (1990).
(There's a lot of detail in McPherson's book on Britain's view of the war. It's also probably the best single volume on the Civil War I've encountered, and well worth reading.)
Great question. First off, it is important to address a key assumption that often comes with US-GB relations in the 19th century: Britain was no longer 'angry' over the outcome of the American War of Independence, thus there was not a majority feeling of schadenfreude once the war started. Indeed, by the mid-nineteenth century, trade with the US had proven to be far more lucrative post-independence than before, particularly due to British reliance on cotton from the US South and Britain's relatively early industrialization. This economic reality was often used as evidence by Radicals and Liberals in Britain to argue for the dismantling of the British Empire (or at least an aggressive process of self-rule and decentralization of power). Thus, American independence was not viewed as an objectively bad thing for Britain - it was contentious, to be sure, but the popular political view of the day was that it was a net positive for British domestic prosperity.
Socially, there was some sympathy amongst the landed elites in Britain with the Confederacy, largely because they saw the wealthy Southern plantation owners as the 'landed aristocracy' of the US. Furthermore, these same landed aristocrats saw themselves in a similar position of being 'forced out' by the rising tide of industrial wealth – the abolition of the Corn Laws in GB, and the shift towards free trade, were fairly recent events in GB, both of which were damaging to the long-term wealth and political power of 'old money' in Britain.
The economic relationship between Britain and the US also played a major role in Britain's views on the Civil War. The reliance on southern cotton to power Britain's textile mills meant that the outcome of the Civil War, and the devastation of the war itself, had the potential to directly affect British industrial prosperity. This is largely why Britain began looking elsewhere, specifically towards Egypt, for their supply of cotton, and also why the British briefly considered throwing their weight behind the Confederacy. While the chance of Britain entering the war was remote, they did offer some assistance to the Confederacy, primarily through the construction and sale of warships to the CSA. An excellent example of this is the postwar lawsuit brought against Britain by the US regarding the commercial damaged caused by the CSS Alabama, a ship built by the British. In short, the US wanted compensation for damages caused by British ships sold to the Confederacy during the Civil War. This behavior is indicative of Britain's views on the Civil War (and really Britain's imperial policy throughout the mid-nineteenth century): disinterest in formal intervention except where absolutely necessary, extreme interest in informal influence via economic soft power and free trade.
Sources:
I don't have anything to add that hasn't already been said, but probably the most definitive work on Britain's involvement in the American Civil War is "A World on Fire: Britain's Crucial Role in the American Civil War", by Amanda Foreman, published in 2010. It talks about public opinion, economic implications of the conflict, and diplomatic relations both with the Union and the Confederacy.
I remember from an askhistorians a week or so ago that the military brass thought it was less a war of true soldiers and more a war of unruly mobs.
At first it seems like pompousness but looking at a comparison of shots fired/hits in concurrent European conflicts shows that they were relatively correct.
Edit: Here it is http://www.reddit.com/r/AskHistorians/comments/yvfvh/what_did_europeans_think_of_the_military/