More specifically, the French government between the Storming of the Bastille and Napoleon's coup d'etat.
How exactly did this government come to power after the monarchy was overthrown? Did it have power over all of France? Was it centralized at all, or was France in a period of anarchy?
The storming of the Bastille didn't change the composition of the government. The head guy in charge was the king, with the National Assembly increasingly making the laws despite him.
The National Constituent Assembly was dissolved in 1791 after their goal of creating a constitution had been fulfilled. It was replaced by the Legislative Assembly, which took over like after any normal election, except with only new deputies, as no one was allowed to run for reelection.
The Legislative Assembly was in power between October 1791 and August 1792, when a second revolution led to the fall of the monarchy. The Legislative Assembly called for new elections, and the National Convention took over in September. France was ruled by the Convention until October 1795. Then the Directory took over, with the Council of Ancients and the Council of Five Hundred as the two chambers of legislature, and the Executive Directory.
So yes, there was a centralized power in France throughout the Revolution. However, at times it lacked control over significant areas that were in rebellion. For example Lyon, Marseilles and Bordeaux, where control was lost in 1793 and regained after a civil war.