Modified from an earlier answer

Although his partisans often invoke Napoleon's Jewish policies and his ending of antisemitic legislation as an example of his enlightened despotism, the reality behind Napoleon's Jewish policies is more prosaic. In short, Napoleon acted upon existing Revolutionary and Republic legislation on the Jews and carried them through to their logical conclusion with a Napoleonic gloss.

In the case of the French Revolution and the subsequent Napoleonic Era, there was a strong tendency of the state to sweep aside many of the restrictive laws against Jews. This was of a piece with the Revolution's Enlightenment heritage which emphasized a rational legal and social order. The Enlightenment operated under a schema which postulated that disorder and crimes originated from imperfections within society. If Jews were associated with such activities like usury and swindling, then eliminating restrictive laws would end that activity. Very few Revolutionaries would turn the mirror on themselves and conclude that the Jews' negative image was the result of gentiles' own antisemitism; they took this antisemitic image of the Jew as an exploiter of gentiles as a given. The Revolution fostered the elimination of corporate privileges of the Jewish communities, largely clustered in the borders at Alasace, and made the tentative first steps towards accepting Jews as full citizens of France. The moderate liberal Stanislas Clermont-Tonnerre would argue in the National Assembly that:

We should refuse the Jews any rights as a separate nation, but grant them the same as individuals. They must not make in the State either a distinctive corps or an order. They must become individual citizens.

Similarly, the revolutionary firebrand Henri Grégoire published a pamphlet in October 1789 favoring Jewish emancipation. Grégoire's motions for Jewish emancipation received tentative support from other political actors like Mirabeau and Robespierre. In the latter case, the Jacobin leader saw the lack of Jewish integration into French culture as baleful legacy of the monarchical regime:

The vices of the Jews have their roots in the degradation into which you have plunged them. They will be good when they see some benefit in being so.

Yet the chaos of the Revolution's later stages meant that emancipation, which passed in the National Assembly in 27 September 1791, would be highly patchwork and incomplete.

It was up to Napoleon to complete this undone task. The Emperor shared much in common with Robespierre with regards to French Jews. In an 1806 letter to the Minister of the Interior, Napoleon would assert that the integrating the Jews into French society through social institutions like the army would eliminate

the tendency of Jewish people to practice a very great number of activities that are harmful to civilization and to public order in society in all the countries of the world.

In the few state meetings that delved into the Jewish question, Napoleon would contend that the ancien regime's classification of the Jews as a separate corporate entity only enabled the Jews' own worst vices. Like his Revolutionary predecessors, Napoleon did not question his own antisemitic precepts about Jews.

Napoleon was in many respects the culmination of the Revolution's Jewish policy. Although the National Assembly had granted Jews full citizenship rights towards the end of its existence, the new revolutionary state also sought to subordinate and systematize Jewish religious and social institutions. Napoleon, who was a systematizer par excellence, convened a council of Jewish elders, the Grand Sanhedrin, to declare that Jewish conscription was religiously valid. This followed other governmental attempts to regulate its Jewish community, ranging from religious dress or the state control over rabbis. Napoleon ushered in the French government's century-long attempt to subordinate religion to the state oversight and control. Although the French state also tried to subordinate the Catholic church to the state as well, Napoleonic and post-Napoleonic approaches to Judaism often treated Jews as community that needed to be corralled into "correct" behavior. Berr Isaac Berr, the Alsatian Jewish delegate to the National Assembly, perceptively analyzed the antisemitism latent in these emancipatory initiatives in an 1806 pamphlet:

Hardly had the decree which granted Jews the title of citizen been handed down than they were reproached with not yet having become worthy as a group for this great favor.

Berr's pamphlet prophesied many of the problems that Jews in France would face after the Revolutionary Era. The systematizing and acculturating processes did turn a number of Jews into Frenchmen over the long term, but the larger acceptance of social antisemitism within the French gentile community acted as a counterweight to total integration. Sometimes this discrepancy between legal and social realities would explode, as in the case of the Dreyfus Affair.

Sources

Aston, Nigel. Religion and Revolution in France, 1780-1804. Washington, D. C.: CUA Press, 2000.

Hyman, Paula. The Jews of Modern France. Berkeley: University of California Press, 1998.

Tozzi, Christopher. "Jews, Soldiering, and Citizenship in Revolutionary and Napoleonic France." The Journal of Modern History 86, no. 2 (2014): 233-257.