How would one attempt a Marxist reading of Jane Austen's Pride and Prejudice?

First, I would point you to this excellent answer elsewhere :

I would add that any Marxist reading of the book would also necessarily consider Marx's concept of alienation. Marx saw alienation as a basic estrangement of the individual, as a person, from his role in society. In a capitalist society, according to Marx, the worker, who actually through his labor produces wealth, is nevertheless alienated from the thing that he produces, since...

First, I would point you to this excellent answer elsewhere :

I would add that any Marxist reading of the book would also necessarily consider Marx's concept of alienation. Marx saw alienation as a basic estrangement of the individual, as a person, from his role in society. In a capitalist society, according to Marx, the worker, who actually through his labor produces wealth, is nevertheless alienated from the thing that he produces, since the capitalist owns the means of production. The worker is only a cog in the capitalist machine.

Characters like Lady Catherine or Mr Darcy might seem at first to have little to do with the production of wealth; in fact, the definition of a "gentleman" is someone who lives off the labor of others. However, in my view Darcy and the rest are as much victims of capitalist alienation, as much "cogs in the machine," as any worker. By this I mean not only the property law that would strip Mr Bennet of his estate on his death, but the rigid social rules that determine rank and subordinate personal values or inclination to status and money. Almost every character in the book is reified in this way, that is, their identity as a person is replaced, or erased, by their position in society, as determined by their wealth.

One possible avenue such a Marxist reading could pursue would be to examine the attitudes of characters to this social hegemony. Charlotte, for example, makes a calculated choice in marrying Mr Collins in order to obtain her own home and some measure of independence. Bingley is another interesting study, a character who is enchanted by Jane, but then is influenced to drop her because of her lower social standing.

In fact, Bingley's story, and Darcy's decision to influence him against Jane, brings to light an important thematic element in Pride and Prejudice, which is that Austen is always cognizant of a counterforce in her characters that is at odds with the bonds of social alienation. Bingley does love Jane after all, and in the end his affection for her overcomes whatever scruples he might have had in marrying her. Darcy is an even better example; his first disastrous proposal to Elizabeth fails precisely because he is conscious of the cost to him personally of breaking with society and proposing to Elizabeth. And her acceptance of him comes only when she perceives that his reasons for acting against Mr Wickham are derived not from his pride but from an honest concern and love for his sister, Georgiana.

It is hard to argue that Austen is somehow trying to overturn social hegemony through the marriage plot. After all, the result of every Austen novel is the marriage of the protagonist, a kind of happy entrance into the very bonds of social custom characters like Elizabeth flout before they are married. But clearly Austen sees that the right basis for any social contract is not the alienation or subjection of wife to husband, or worker to production, but in honest and forthright human feeling.