The majority of books I have read, and what I have been taught in school, invariably focussed on the French leadership; the various assemblies and committees that helped guide the revolution. I was curious about how Marxism can be used to interpret the events of the French Revolution from 1789-93 from the perspective of the people, and are there any books in particular you would recommend to help understand the French Revolution from this viewpoint?
The French Revolution can be and was definitely extensively interpreted from a Marxist viewpoint. This path of description was at the forefront of French historiography of the period from the beginning of the XX^th century well until the 1970s at least. The description is not "from the perspective of the people" (this is, arguably, not the core point of dialectic materialism) but showcases the French Revolution as the Ur-Beispiel of Revolution resulting of a class struggle (Klassenkampf) and an heterogeneity between the social structure and the relations of production (Produktionsverhältnisse). As the French economy was increasingly ready for transition from a feudal mode of production to a capitalist mode of production, after a primitive accumulation of capital owing, in part, to the fermiers généraux, and to the development of international commerce (think about the indiennes dressing gowns of M. Jourdain in Molière's Bourgeois Gentilhomme or the merchant fortunes of Nantes and Bordeaux in the second half of the XVIIIth century), which allowed the emergence of a capitalist class.
The mismatch between the organization of productive forces and the organization of power in society is well illustrated by the famous formula of the Abbé Siéyès in his 1789 Pamphlet "What is the Tiers Etat ?" :
« Qu'est-ce que le Tiers-État ? Le plan de cet Écrit est assez simple. Nous avons trois questions à nous faire. 1º Qu’est-ce que le Tiers-État ? Tout. 2º Qu’a-t-il été jusqu’à présent dans l’ordre politique ? Rien. 3º Que demande-t-il ? À y devenir quelque chose. »
« What is the Tiers Etat. The structure for this paper is short and simple : we have 3 questions to answer : 1º What is the Tiers Etat ? Everything. 2º What has it been so far in the political order ? Nothing. 3º What does it demand ? To be something. »
This historiographical courant was (in part) superseded by the works of the revisionist school, and in particular the works of F. Furet (La Révolution Française, 1965) and the works (later) of M. Gauchet (La Révolution des droits de l'homme, 1989)
The main authors of the Marxist school on the French Revolution (in France) would be :
Lefebvre, and in particular Quatre-Vingt-Neuf (1939), with a very orthodox marxist train of analysis of the summer of 1789 events (step 1 the Revolution of notables and aristocrats, step 2, the Burgher revolution, step 3, the people's revolution with the storming of the Bastille). He was one of the early proponents of the description of the Revolution as a capitalist class toppling over a feudal aristocracy. If you like "history seen from the people's perspective", Les Paysans du Nord pendant la Révolution française (1924) is a good description of what happened to the peasant population in a relatively developped part of rural France in that period.
His pupils, A. Soboul (Les sans-culottes, 1968) and G. Rudé (The Crowd in History. A Study of Popular Disturbances in France and England, 1730–1848). The second one was English and published in English so he could be a good place to start, and the style will probably be less dated than Lefebvre.
There is still a living marxist school in France, with some specialists trying to identify in the Parisian sans-culottes movements the emergence of very early "worker's" organizations, announcing the later events in the XIXth century. A good example (specializing in the analysis of G. Babeuf and his movement) would be Schiappa (father of the current Secretary of State for Women's rights).