Marxist literary criticism attempts to read texts for discovery of (1) reference, direct or indirect, to oppression of the working classes by the privileged class and for the disclosure of (2) the economic situation at the time the text was written. What this means in terms of Pride and Prejudice is that a Marxist critic would read to find evidence of oppressed alienation of workers, for example, domestic staff, and for indications of the economic conditions in the text and during Austen's writing years, spanning from the late 1790s to the early 1810s, as she finished writing Pride and Prejudice in 1798 while it was not until 1813 that it found a publisher (recall that the text was not modified to reflect socio-economic changes, if any, that may have occurred in the 15 year span).
The analytical tools used by a Marxist critic in a Marxist reading are examination of the text for indications of economic oppression of workers; alienation (estrangement) of workers from their creative selves, from other workers, from the products they make, and from the creative process that their labor advances but that is not under their control; economic exploitation by the upper classes resulting in conflict between classes.
Other analytical tools used are examination of the text for indications--related to the text and related to the author's own time period--of the economic base and superstructure, ideology and hegemony, and reification of workers. The economic base is the economic principles that establish the social and cultural order of things, which is called the superstructure and which includes such as religion, education, law, and art. Ideology is a belief shared by all (or most) in a society about the way things should be that grows up out of the superstructure, which is itself determined by the economic base. Hegemony is a collection of such ideological beliefs. Reification of workers is their objectification so that they come to be considered commodities available for market exchange, as in situations where there is great competition for available jobs: people aren't hired, rather commodities are hired because they are readily exchanged in the market place.
An example of how these Marxist critical principles would be applied to a Marxist reading of Pride and Prejudice might easily be illustrated by looking in the text for any signs of domestic servants. Despite the (misguided) portrayals by Hollywood of the Bennets' life, we know, from Austen's text, that Bennet, though having married beneath his social class, is a country gentleman of the upper class who has (rather had) a large income and ample estate. Mrs. Bennet is therefore a lady who, we are told in the text, employs a cook (whereas the Lucas's have no cook). We know that with five daughters, a lady of Mrs. Bennet's stature will certainly not do the housework and laundry etc herself. We know too that the Bingley's are even more elegant ladies with even more wealth (theirs not being wasted yet) and that Bingley and certainly Darcy would not expect less than a full quorum of servants.
"... he came down on Monday in a chaise and four to see the place, and ... he is to take possession before Michaelmas, and some of his servants are to be in the house by the end of next week."
We know too that Pemberley and Rosings are no less endowed with servants than Netherfield, yet these are virtually the only two references we read to servants.
A Marxist reading would analyze this as oppression, alienation and reification. The servant workers are so oppressed, thus dehumanized, that they do not even merit mention except insofar as their mention is used for characterization (Mrs. Bennet) or to forward the plot (Bingley comes by Michaelmas). This oppression, in the text and in Austen's real economic times, alienates servant workers from their own identities and alienates them from others. In addition, the upper classes are alienated from their own humanity by the reification of people who become commodities that perform or not and can be exchanged in the market place after being oppressed, alienated and dehumanized.
eNotes offers a study guide on Pride and Prejudice which can be found in its entirety at the link below. The study guide includes a summary, analysis of character, an explanation of themes, and links to critical essays.