How does Napoleon continue the revolution?

Napoleon continued the French Revolution first by resisting the external threats to the Revolution. Namely Great Britian and other European states who felt threatened by the Revolution and who sought to replace the new government by putting another Bourbon king on the throne. He also continued the Revolution by supporting the government itself. As he gained power he continued the reforms to the government and law that the Rvolution had started, and did so in such a just manner that his model would be used throughout Europe in the future. He also made the French government a meritocracy, where it was one's ability that determined to what rank and position you could rise, rather than the accident of birth.

On the betrayal side, Napoleon's biggest action was in accepting the role of Emperor. He in effect turned back the clock to claim a royal title in a nation that had rejected the idea of royalty. He would also go on to award noble titles to his best and loyalest supporters, as well as placing family members on European thrones in nations he had beaten. He also betrayed the Revolution in his conquests of other nations. This was partially a defensive measure against the intrigues of those nations, and partially an attempt to fulfill his own ambitions to earn the title he had bestowed on himself. But in doing so he condemned a generation of Frenchmen and youth to a life of constant warfare, left the economy of France hanging precariously in the balance as it tried to support his war needs and fight Brithish embargoes, and bled all of Europe of men.

Most likely Machiavelli would approve of Napoleon. Napoleon had ambition, nerve, ability, and a willingness to do what must be done to gain and maintain his position. Napoleon did so and managed to retain the love and loyalty of most of France's citizens throughout most of his reign.