How would I classify the characters of the novel Emma by Jane Austen according to their social rank or class stratum?
Since our society is vastly different from 19th century English society, you need an external resource in order to thoroughly classify Austen's characters according to their rank. You can tell by yourself what the ranks of some are from general knowledge or from remarks or hints in the text, but for a thorough classification, you need more than general 21st century knowledge.
Displayed online on JaneAusten'sWorld.com is a replica of the 1814 "Map of English Society" document. It details the class divisions. It is taken from Patrick Colquhoun's A Treatise on the Wealth, Power, and Resources of the English Empire, 1814. It displays and defines the seven plus one recognized classes in English society for the period during which Austen wrote Emma.
Looking at the Map, you'll see that no character in Emma is of the "Highest Orders." There are no royalty, lords, Officers of State or "peers above the degree of baronet" in this novel. There are three families in the Second Class as this class includes "country gentlemen" and "others with large incomes." Second Class may be attained through merit resulting in wealth while the Highest Order must be attained through heredity or through royal "creation" (royal designation of a new title above baronet: England's Princess Kate was created Duchess of Cambridge by Elizabeth II so that she might wed Prince William).
Mr. Knightley, the Churchills (including the adopted Frank Churchill) and the Woodhouses are in the Second Class as being country gentlemen and/or having large incomes. Miss Bates and her mother were of some higher order than they are during the chronology of the narrative, but we are never given specifics. Thus, while they may have been Second Class, they are now most likely Fourth Class, "persons of moderate income," as they are neither workers nor paupers.
Harriet Smith appears to be in that same class, Fourth Class, since she neither works nor is left wanting for any material thing due to a "moderate income" from her father. Mr. Elton is a Third Class clergyman since (1) he is of a high enough status to socialize with and be called friend by Mr. Knightley and (2) is the clergyman for Highbury, a wealthy town with several prominent wealthy citizens besides Knightley, Woodhouse and Weston. Mrs. Elton is in the Third Class in her own right since she has twenty thousand pounds in her name before marrying Mr. Elton. While twenty thousand wasn't enough to be "large income," it was certainly a nice income and quite enough to confirm their solid position in Third Class.
Using this information and the Map, you can determine the social class of other interesting characters, like Knightley's brother, the Campbells, the Westons, Jane Fairfax, Farmer Martin, the Coles, Mr. Perry and Mr. Cox.