Modified from an earlier answer

One of the hallmarks of the Napoleonic tax administration was that it introduced a degree of efficiency and systematization that meant the Empire could count upon a steady stream of revenue. The recent memory of Revolutionary hyperinflation inhibited more experimental and flexible approaches to finance. For example, one of the achievements of Napoleon that his proponents frequently trumpet, the Banque de France, played a relatively marginal role in governmental finance. Napoleon's aversion to short-term loans meant that ensured that the Banque's contributions to French state financing was never more than 10 percent of total expenditures. The state's commitment to metallism, another legacy born out of a painful Revolutionary experience, meant that the Banque adapted the "Palmer Rule" in which a third of its notes in circulation would be backed by metallic currency. The main sources of war revenue for France would be the systems of direct and indirect taxation from domestic sources and contributions from defeated or allied states.

There is a strong line of continuity between the tax policies of the later Revolutionary governments and the Napoleonic Consulate and Empire. Although the Directory resorted to paper currency and suffered from a repeat of hyperinflation, there was a strong corpus of administrators within the French state who advocated a more stable French economy and logical reforms. The Directory relied upon a series of land taxes, contributions, and later, indirect taxes to finance the French government. The main impact of Napoleon on the French tax structure was that his characteristic focus upon systematization and efficiency, coupled with the promotion of able servitors that were already advocates of reform. One of the unsung administrators of the Napoleonic period was Martin-Michel-Charles Gaudin, Napoleon's Minister of Finance from 1799-1814, and 1815.

Gaudin directed an overhaul of the la contribution fonciere(land tax), an assessment based upon agricultural income. This form of direct taxation made up to three-quarters of all taxes derived from direct taxation for most of the Napoleonic period. The Ministry of Finance enacted systemic cadastral surveys in 1802, and expanded them into a massive project to catalog the whole of France in 1807. The grand cadastral surveys assessed France parcel by parcel, judging soil quality, buildings, and other factors to predict the ideal agricultural value of the land. The surveys rendered this particular tax burden more equitable, but also expanded both state power and control.

Another form of direct taxation in the Napoleonic period was la contribution personelle-mobiliere (taxes on personal or industrial incomes), which was a legacy of the Revolution and mostly hit towns. This somewhat progressive tax had a fixed sum, plus a variable amount based upon external signs of wealth like chimneys or the number of servants. In keeping with its principles of systematic reform and efficiency, Napoleon overhauled this highly arbitrary system between 1803-04. Its main replacement was the droits d'octroi which was a levy upon all goods entering into a town. There were also various taxes licensing trades and services and upon items like doors and windows.

Although these taxes introduced a steady and reliable stream of income to the French government, they could not cover all of the expenses of the Napoleonic state. Napoleon reintroduced various forms of indirect taxes to make up for this shortfall. These droits reunis were levied upon tobacco, playing cards, alcohol, and salt became increasingly important for French finance from 1806 onwards. The revenue from these indirect taxes increased some fourfold between 1806 and 1812. Again, in keeping with Napoleonic centralization, the state instituted a central excise office to enforce these indirect taxes. Additionally, the French government created a state monopoly on tobacco in 1810. Collectively, these excise taxes made up a quarter of France's tax revenue in 1813.

Yet even Napoleonic efficiency could not keep pace with the growing costs of France's wars. The Napoleonic fiscal system became increasingly dependent upon making war pay for war. From 1806 to 1814, non-French states bore more than half of Napoleon's military expenses. The quartering of troops in Italy, Central Europe, and Spain displaced some of the defense burden upon Napoleon's allies and occupied states. Indemnities upon defeated powers became a normal means of the French to cover their budgetary deficits. Between 1806 and 1812, Prussia had to provide France with somewhere between 470 and 514 million Francs. To put that figure in perspective, the Banque's contribution to the French budget peaked in 1805 with a figure of 80 million Francs and provided the campaign of 1805 60 million Francs. Even Napoleon's allies were not immune from this system of contribution. Tax levels in the southern German states doubled from their 1806 levels, but this was often not enough to keep pace with fiscal demands. The Rheinbund states had to engage in a tricky shell-game of forced bonds purchases to pay for their upkeep. In the Kingdom of Westphalia, 1808 bonds that were due in 1811 were paid off with more bonds. The Hanseatic towns of the northern German coast, officially annexed to the French Empire in 1810, had the French system of indirect and direct taxes imposed, but imperial authorities were more diligent at squeezing out whatever wealth they could get out of these German cities than those in France. Due to their proximity to the Central European battlefields, it was more urgent to collect imperial taxes from these areas. In Hamburg, between 1810-13, approximately a third of all the city's taxes went to outfitting the French military in Germany. The city's support infrastructure for Hamburg's indigent correspondingly suffered as orphanages and hospitals closed up shop.

While this system of contributions and making war pay for war was not unprecedented in European history, but it added another onerous burden upon the other middling and great powers of Europe and made it much harder for Napoleon to cement a lasting peace. The contributions system created a persistent grumbling about French exploitation and undercut Napoleon's attempts to forge a working alliance with these small states. The disaster of 1812 made pushed this precarious economic system off a cliff as Napoleon had to wring yet more revenue from his allies, and increasingly, from metropolitan France. Despite his reservations about short-term financing, Napoleon assented to sale of short-term bonds within France and expanded the various indirect tax systems. By 1813, many of the states in central Europe were approaching physical insolvency and had an inability to pay their civil servants.

One solution for these European states to meet French demands was to implement French fiscal methods. In the Kingdom of Italy, Giuseppe Prina played the role of Gaudin by strengthening local fiscal institutions and conducting cadastral surveys to render the system more efficient. Although a mob murdered Prina in 1814, his methods lived on. During the post-Napoleonic period, many European states continued the trend of systematizing and streamlining tax revenue through the increase in state powers of regulation and control. The utility of a steady revenue stream and increase in state power often transected ideological barriers in ways that other legacies of the French Revolution, such as nationalism or popular sovereignty, did not. Even Piedmont-Sardinia, one of the most reactionary of the post-1815 governments, had no problem embracing the French-created system of tax inspectorates and gendarmes to enforce order as essential component for the operation of the state. In France, the Bourbon and Orleanist governments continued Gaudin's cadastral survey would not be finished until the 1820s, but was one of the bedrocks of French tax policies up to the First World War. Napoleonic occupation created a need for an overhaul of revenue collection within continental Europe and this new found efficiency outlasted Napoleon. Although it lacks the drama of the corps system of Napoleon or other elements of martial glory, the reform of tax codes was a persistent legacy of the Napoleonic period and impacted the daily lives of Europeans for the coming century.

Sources

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Bergeron, Louis. France under Napoleon. Princeton, N.J.: Princeton University Press, 1981.

Bordo, Michael D., and Eugene N. White. "A tale of two currencies: British and French finance during the Napoleonic Wars." The Journal of Economic History 51, no. 02 (1991): 303-316.

Chadha, Jagjit S., and Elisa Newby. 'Midas, transmuting all, into paper': the Bank of England and the Banque de France during the Napoleonic Wars. No. 1315. School of Economics Discussion Papers, 2013.

Ellis, Geoffrey James. The Napoleonic Empire. Houndmills, Basingstoke, Hampshire: Palgrave Macmillan, 2003.

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The funding for Napoleon's campaigns differed during his time as a General, First Consul, and then Emperor. Since you mentioned that the French government was near bankrupt during the revolution you might be interested on how he funded his army during the French Revolutionary Wars.

When Napoleon took command of the Armée d'Italie in 1796 they were indeed lacking in funds(among other things). The army he inherited was supposed to be 43,000 strong but was actually closer to 30,000. They only had 60 guns, many cavalrymen had no mounts, a lot of the infantry didn't have shoes or uniforms, and all were unpaid.

His first proclamation to the army was, "Soldiers, you are naked, ill fed... But rich provinces and great towns will soon be in your power, and in them you will find honor, glory, and wealth. Soldiers of Italy! Will you be wanting in courage and steadfastness?"

This is exactly how Napoleon funded his first major campaign. Promising his soldiers loot encouraged them, and his success in battle allowed his army to loot the countryside. The House of Savoy, the Habsburgs, and all the small states had thousands of churches and other religious locations with loot. Works of art and religious vessels of gold and silver were all liberated from the churches and loaded on his supply carts for "safe keeping."

The soldiers got their petty loot and pay, while Napoleon transferred much of the loot to both his own account and to the government in Paris. It is estimated that Napoleon's forces took over $50 million in loot during the Italian Campaign.

He also had his eyes on the riches of the Orient. The campaign in Egypt was undertaken in the guise of providing valuable scientific and historical knowledge to the new French Republic. However, Napoleon saw the spoils to be taken from North Africa, the Levant and areas controlled by the Turkish Empire. All this eventually leading to India where he would join with the Marathas and Tipu Sahib and overthrow British rule there.

The Directory agree mainly because Napoleon had become incredibly popular following his successes in Italy. Fearing a power grab, they could possibly have him involved in years of war thousands of miles away. They also required that Napoleon fiance the invasion himself. He had Berthier raid the Vatican's treasury, Brune took the entire Swiss reserve, and Jourbert was sent to the Dutch forcing them to pay as well. He was able to raise 10+ million francs for the Armée d'Orient. Like his campaign in Italy, the looting of religious sights continuted. After convincing the Knights of Malta to surrender, he looted their treasury, churches, and convents.

TL;DR - Promised soldiers loot, won battles, ransacked the countryside.

Napoleon: A Life - Paul Johnson

Napoleon: A Concise Biography - David Bell