In the initial introduction to Perkins, Doyle styles him as the chauffeur when the narrator says, "asking Perkins, my chauffeur, how [the car] ran." This immediately sets up the class difference between them while establishing the relationship of master and servant. Doyle has already established the setting as around London, England, thus refining the relationship between them to reflect the particulars of the British institution of domestic servants.
Some distinctive points in Perkins's characterization mark him as exceedingly competent, courageous and devoted. When the narrator wants to take the wheel of his freshly delivered new car, his "new thirty-horse-power Robur," Perkins--a competent expert with cars, as any good chauffeur is--replies that the car's transmission gears are not like that on the old car and suggests that he himself should therefore drive: "’The gears are not the same,’ [Perkins] said. ‘Perhaps, sir, I had better drive.’" He calls the narrator “sir” to denote their class differences and their master and servant relationship.
Later when the car is rushing down the hill out of control, Perkins courageously takes momentary control of the situation by taking hold of the steering wheel and suggesting the narrator jump to safety since they "can never get round that curve." Perkins urges his master to safety by saying, "Better jump, sir." When the narrator replies, "No ... I'll stick it out. You can jump if you like," Perkins shows his devotion to his master--despite his conviction that they will not make the next curve--by saying, "I’ll stick it with you, sir."