Strictly speaking, Karl Marx cannot be considered a literary critic. His concepts, however, have proved beneficial to literary criticism and allow for original, new interpretations of literary works. Discuss.
Yes, Karl Marx was not a literary critic or a literary theorist. He was a philosopher of history. His theory of historical materialism shows how the production of material goods is the driving force of history. Marx also showed how this production of material goods interacts with, and causes, social and cultural aspects of society. So, the way a society operates informs the beliefs of that society. Consequently, artists and writers will necessarily be informed by those beliefs.
A literary critic who focuses on class tensions, economics, and historical context is focusing on material factors (historical materialism). These factors are the "base" which give rise to ideas, cultures, and ideologies. These ideas and ideologies are the "superstructure." Therefore, the base (actual work and social relations) informs the ideas we have about life. In Critique of Political Economy, Marx writes:
It is not the consciousness of men that determines their being, but, on the contrary, their social being that determines their consciousness.
Under this belief, if a person or society wants to change the way they think and the things they believe, they must change the material conditions of society.
As long as the mode of production is sustained, a certain ideology will continue to have influence. When a ruling class sustains its domination of the lower classes, it does so by continuing the modes of production (which continues to condition/reestablish the ruling ideology). This domination is called "hegemony." Marxist critic Louis Althusser designated "ideological state apparatuses" as institutions that recondition the hegemonic rule. These can include political parties, unions, the media, courts, schools, churches. He also designated "repressive state apparatuses" such as the police, military, and government. These apparatuses indoctrinate and pacify (more than force) members of society to accept class distinctions. A Marxist critic dissects the ways a hegemonic order structures society and its people. As a result, artists and writers are subject to these same influences. So, the Marxist critic looks at how writers are indoctrinated by these influences or how the writer undermines these influences and points out the problems within.
Since Marxist criticism deals largely with social and economic structures, Feminists and Race Theorists have found it to be a useful supplement to their own theories because they look at the ways women and minorities have been conditioned to live, how they've been repressed, and in some cases, how they've been brainwashed to accept their status in society.
One of the things Marxist criticism (as well as Feminism, Race Studies and New Historicism) has done is to suggest that it may be irresponsible or impossible to study a literary work without considering the social and political realities from which it was written. A Marxist critic would challenge a text which blindly accepts the hegemonic order. The idea of roles (gender, economic, social) is also something a Marxist critic might challenge because an ideology is only sustained if people continue to accept their roles in society. The critique of roles has become a more common criticism in Feminism, Race Theory, Gender Theory, and Cultural Studies.