No external country backed either the Union or the Confederacy, despite Confederate hopes that they would find allies. There were various possibilities of intervention, but none were very strong, and none happened.

The British:

The Confederacy had high hopes that Britain might come in to the war on their side. They thought that cutting off the cotton, on which the textile mills of northern England depended, would bring Britain in to the war on the Confederate side to restore supplies.

The Confederates underestimated the moral opposition to slavery in Britain, which made support of the Confederate cause by the British unlikely.

One of the most principled and courageous stands against slavery and in favor of the Union came from the mill workers of Lancashire in England. It deserves to be better known in the USA than it is.

In 1860 there were 2,650 cotton mills in Lancashire, employing 440,000 people. As the Union blockade, (and a Confederate embargo, aimed at bringing Britain into the war on their side) cut off the cotton, the mills shut down and the workers lost their jobs.

Despite hardship and starvation, in 1862 a meeting of the increasingly desperate mill workers still resolved that they would support the Union and suffer if it led to the ending of slavery.

(Source: Sir Arthur Arnold, “Cotton Famine 1961-1864”, 1864)

They sent this letter to President Lincoln:

"... the vast progress which you have made in the short space of twenty months fills us with hope that every stain on your freedom will shortly be removed, and that the erasure of that foul blot on civilisation and Christianity – chattel slavery – during your presidency, will cause the name of Abraham Lincoln to be honoured and revered by posterity. We are certain that such a glorious consummation will cement Great Britain and the United States in close and enduring regards."

—Public Meeting, Free Trade Hall, Manchester, 31 December 1862. (Source: http://www.manchester.com/features/manchesters.php)

Lincoln sent the following letter back:

"Under the circumstances I cannot but regard your decisive utterances on the question as an instance of sublime Christian heroism which has not been surpassed in any age or in any country. It is indeed an energetic and re-inspiring assurance of the inherent truth and of the ultimate and universal triumph of justice, humanity and freedom.

I hail this interchange of sentiments, therefore, as an augury that, whatever else may happen, whatever misfortune may befall your country or my own, the peace and friendship which now exists between the two nations will be, as it shall be my desire to make them, perpetual."

—Abraham Lincoln, 19 January 1863 (Source:http://www.bbc.co.uk/news/world-21057494)

Lincoln sent more than just kind words. On 9 February 1863 the ship George Griswold, docked in Liverpool. It was sent by Lincoln, aided by the citizens of Philadelphia and New York, and carried bacon, bread, rice, corn, and 15,000 barrels of flour to relieve the suffering of the Lancashire mill workers.

(Source:http://revealinghistories.org.uk/tpl/uploads/Education-card-4.2.pdf)

Although some elements of British society might have been tempted to support the Confederacy, and although Union bungling (in the Trent incident) came close to enraging the British and bringing them in on the Confederate side, Prince Albert personally got involved to calm things down and keep Britain neutral. 440,000 principled and courageous mill workers were part of the reason why.

The French:

The Confederates also hoped that the French might support them. The French, however, had a different strategy, which they thought might give them more benefits and carry fewer risks.

In 1861, while the US was pre-occupied by the Civil War (and unlikely to raise Monroe Doctrine objections), France invaded Mexico. In 1864 they installed Maximilian of Austria as Emperor of Mexico. The French never managed to gain control of all of Mexico. National opposition forces, led by Benito Juarez, even though pushed into marginal areas, managed to keep fighting.

When the Civil War ended, the US began to send aid to Juarez, and the French withdrew their troops in 1866. Maximillian was captured and executed in 1867.

The French were too involved in their Mexican adventure to really contemplate getting involved in the Civil War.

Russia:

The Union thought that they had at least the moral support of Russia during the civil war.

Russian reaction to and involvement in the US Civil War is an interesting story.

The Russians actually sent a considerable portion of their navy to the USA in 1863. Russian battle fleets spent almost a year in New York and San Francisco during the Civil War. The Union took this as a sign of Russian support for their cause, and there were a lot of receptions and parties for the Russian officers.

To some extent, the Russians did support the Union. At one point, the Russian admiral in San Francisco issued standing orders to his battleships to engage and destroy any Confederate raiders entering San Francisco Bay. Later, after clarifications from home, these orders were altered to, maintain neutrality unless a Confederate raider actually bombarded San Francisco. Then, on "humanitarian grounds" the Russian battleships were ordered to blow the transgressing hypothetical Confederate raider out of the water.

No Confederate raider actually entered San Francisco Bay.

In actual fact, and contrary to American beliefs, the Russian presence in US harbors was not caused by a desire to show support for the Union. It was part of a Russian battle plan for a feared war against Britain. The real Russian reasons for their fleet deployment only came to light 50 years after the visit, when the Tsar's secret orders to the fleet were revealed in the Russian archives.

In 1863, there was an insurrection in the part of Poland ruled by Russia. Russia felt it likely, almost inevitable, that Britain and France would go to war in support of Poland (Russia had recently fought the Crimean War against Britain and France from 1853-1856.)

Russian admiral Krabbe proposed a bold naval plan to the Tsar. Instead of letting the inferior Russian navy be uselessly trapped in their ports in the Baltic and Pacific, he proposed sending them to the USA to wait for telegraphic confirmation of war. Then they would burst forth to raid British commerce and bombard British colonies, and contribute to the war effort instead of being uselessly blockaded.

These were the secret orders which the Tsar sent to his fleets. So, they sailed to America, and waited for news of declarations of war. The war never came, and after a year, the Russian fleet sailed home again.

The incident led to considerable goodwill between the USA and Russia. It contributed, a few years later, in 1867, when Russia, still worried about war with Britain, thought it might be better to sell Alaska than lose it in war, to a Russian offer to sell Alaska to the USA.

The offer was accepted.

Source: www.loyno.edu/history/journal/1983-4/delegate.htm

No other powers showed much interest in getting involved in the US Civil War. So, no external powers actually involved themselves in the Civil War to any real extent.

You may read from time to time that the Vatican (or the papacy, depending on the source) recognized the Confederacy. Pius IX did write a letter to Jefferson Davis. In that letter, he did address Davis as the “[I]llustrious and honorable President of the Confederate States of America.” However, the full reality is not nearly so dramatic. Here is the text if you want to read it. In sum, the letter is the equivalent of a form letter. It acknowledges receipt of Davis’ earlier letter, says that it would be really nice if there was peace instead of war, asserts that the pope will pray for peace, and that he will also pray that Davis will be granted grace.

That’s it.

So, why the address as “President?” Basically, the pontiff filled in the blank by looking at the signature of the original letter from Davis. With Italian unification and being a self-proclaimed “prisoner of the Vatican,” Pius IX had much greater concerns much closer to home than the Confederacy. Also, the idea that the pope would somehow back the South in the US Civil War is far-fetched, given the less than ideal way Catholics were treated in that region. So, if you hear someone claim that the pope recognized the Confederacy, refer them to this post so they can learn that the truth was much more banal—the pope just replied to Davis’ letter with bland pleasantries.

Not very close. The Confederacy's main thrust in terms of seeking recognition came through their envoys to Britain and France. James Mason was the Confederacy's "diplomat" in Britain, yet no one from Lord Palmerston's government ever officially received him or recognized him as a legitimate agent of a sovereign nation. To do so would have been the first step towards recognizing the Confederacy, and Lincoln had his own man in England to make sure that didn't happen. The U.S.'s diplomat in Britain, Charles Francis Adams, was instructed to make it clear, in no uncertain terms, that if Britain recognized the CSA, then the U.S. would kick up a fuss (and perhaps start another war). Palmerston's government didn't want to risk losing an ally (albeit a somewhat uneasy one) in the U.S. until they were sure that the Confederacy could and would win the war. This moment never came.

France had even stronger diplomatic relations with the U.S., and the Confederacy's agent there, John Slidell, had even less luck. Neither diplomat ever came close to getting a formal declaration of recognition from England or France, in fact, neither was ever even invited to formally make their case to the government or monarch. (So France and Britain wouldn't even hear them out)

[James McPherson, 'Battle Cry of Freedom'; Bruce Catton, 'Mr. Lincoln's Army'; Shelby Foote: 'The Civil War: A Narrative, vol. 1']