I'm going to focus primarily on Britain and France in this answer, because that's what James Macdonald's A Free Nation Deep in Debt: The Financial Roots of Democracy, the best book on the subject I have close to hand, does. In part, the answer to how Prussia, Austria and Russia paid for the armies they mobilized to fight Napoleon was "Britain," which paid out around £60 million in subsidies to help bankroll the campaigns of various Continental enemies of Napoleon, though that represented only a portion of their total costs that had to be covered with taxes and private debt.

France paid for its share of the Revolutionary and Napoleonic Wars through a few different expedients, of which the two most important were taxes and foreign expropriations. During the Directory's rule in France in 1795 and 1796, France's total tax collections had sunk as low as 50 million francs. By 1810 Napoleon (in part because of decisions made by the Directory that took a while to pay off) was taking in 750 million francs per year in taxes. Part of that was new enforcement of the land tax, which had fallen to almost nothing during the revolutionary chaos but were back up to 240 million francs in 1797 and 300 million in 1799. Another huge factor were a range of indirect taxes, which had been a staple of ancien régime finance but were so hated for how they nickeled-and-dimed the French people that the revolutionaries had abolished them. As the wars continued in the late 1790s, however, "it was only too apparent that no modern state could survive on direct taxes alone, and this cornerstone of revolutionary ideology had to be abandoned." The indirect taxes were added back bit by bit, including tolls on goods brought into cities (1798-9), a tax on wine (1803) and the most hated of them all, the salt tax (1807). These indirect taxes combined were taking in more than 200 million francs per year for the state.

In 1806, Napoleon had annual revenues after debt service of around 500 million francs, which got as high as 600 million francs in 1810; the Bourbons, in contrast, were making do with around 170 million francs on the verge of the Revolution. In British pounds, Napoleon's revenues amounted to anywhere between £20 and 35 million per year, depending on how you factor exchange rates.

Of course, the other major way that the Revolutionary and Napoleonic Wars were funded was forcing other countries to pay for it. The early Revolutionary wars had been paid for by seizing the property of domestic enemies, but when the French armies took to the offensive, they made their external victims pay up, too. For example, in 1795, France invaded the Dutch Republic and then made the defeated Dutch pay 200 million francs for the cost of being "liberated." Austria paid 100 million francs in 1805, and 250 million francs after a defeat in 1809. Prussia's disaster at Jena in 1806 saw it assessed 515 million francs "as punishment for its imprudent attacks."

How much was actually paid and how this all adds up is contested, with figures of total forced payments to France from its enemies over the course of the generation of war given anywhere from 785 million francs to 2 billion francs. As a general rule of thumb, these payments can be seen as paying for the cost of France's overseas armies, especially in conjunction with nonfinancial extortion, such as food or fodder, which was also extreme. Forcing someone to feed your army is in a sense not much different than forcing them to pay you money that you use to go buy food. (Other extortions were more symbolic, such as the priceless art the French carted off to the Louvre.)

Across the Channel, Britain, as stated above, was in the business of paying other European countries, not making them pay. (At least, not until 1815, anyway, when a defeated France was levied with a hefty war indemnity.) It spent the early years of the war trying to finance the fighting the same way it had paid for the Seven Years War and American Revolution: borrowing, The Bank of England sold long-term bonds to finance British war spending, and taxes were raised only to cover the interest on the debt. For example, Britain spent £112 million on the American War of Independence, of which 82% was borrowed. The early wars of the French Revolution, from 1793-1797, cost another £100 million, of which 89% was borrowed. But as the wars dragged on, problems started to emerge. The value of British bonds fell well below their face value, from 97% of par in 1792 to 53.25% in 1796. In 1797 the Bank of England was forced to end the convertibility of banknotes into gold and silver to prevent a liquidity crisis, while the British deficit ballooned.

Thereafter, the British accepted that if they were really going to do this whole war-to-the-death-with-France thing, they were going to have to raise taxes. And raise taxes they did! A host of new indirect taxes were levied on luxury goods such as carriages and hair powder, bringing in £2 million per year. Land taxes doubled, excise taxes went up 2.5 times, customs revenues almost tripled. On top of all of that, Britain introduced an income tax: 10% of income, with people below £150 in annual income partially exempt and those below £50 completely exempt. This wasn't assessed only on the wealthy — a male servant at the time might earn anywhere from £20 to £60 per year but the full tax only hit the well-to-do. (By contrast, the Bennet family in Pride & Prejudice earned about £2,000 per year, while the wealthy Mr. Darcy had £10,000.) This tax brought in £6.5 million its first year, £12 million its next (as collection became more efficient), and nearly £16 million by 1813. That's more than the entire pre-1789 budget, and nearly as much from this one tax alone as Napoleon's entire revenues (which, as noted above, were between £20 and £35 million).

This tax, one should note, was incredibly unpopular; Macdonald quotes an 1816 letter to the Times criticizing "the despotic spirit of the inquisitorial impost" and notes that this view as "almost universally held" among wealthy Britons. It wasn't just the cost, though this was bad enough, but also the fact that the government needed to know how much you were earning!

All told British taxation rose by more than 50 percent, from around 13 percent of GNP before the wars to 20 percent afterwards. It also rose in relative terms vis-à-vis France, which had collected half of Britain's tax burden before 1789 and now was just one-third, despite vastly increasing its own taxes during this same time!

Of course, all this taxation was paired with (and paid for) extensive borrowing, as well, about £470 million in all on top of an existing pre-war debt of more than £300 million (seen then as dangerously unsustainable!). In all around half the cost of the Napoleonic Wars were paid for via borrowing, much lower than Britain's major 18th Century wars.

My primary source here is: Macdonald, James. A Free Nation Deep in Debt: The Financial Roots of Democracy. Princeton: Princeton University Press, 2006: 272-346 and especially 327-346.

Additional figures from: Silberling, Norman J. "Financial and Monetary Policy of Great Britain During the Napoleonic Wars." The Quarterly Journal of Economics 38, no. 2 (1924): 214-33. http://www.jstor.org/stable/1884011.