When Darcy says to Elizabeth, "We neither of us perform to strangers," he is referring to the fact that both characters have a tendency to be extremely direct and straightforward with their feelings and opinions, regardless of the impression of themselves they leave upon others. Most people might be more reserved, careful, and even showy among people they are just getting to know, but neither Darcy nor Elizabeth would be willing to do this.
One example of Darcy's directness can be seen in the moment when Jane is taken ill at Netherfield and Elizabeth goes to stay while Jane recovers. During one of the post-dinner conversations, Darcy makes what can be considered a very arrogant remark concerning the accomplishments of women. While Bingley praises all women as being extremely accomplished, Darcy says, he "cannot boast of knowing more than half a dozen, in the whole range of my acquaintance, that are really accomplished (Ch. 8). He further describes that a truly accomplished woman must be able to play music, sing, dance, draw, know all of the modern languages, as well as "possess a certain something in her air" in order to be considered by him as truly accomplished (Ch. 8). Elizabeth is rightly so shocked by such a harsh criticism of women that she says it is no wonder he only knows six accomplished women. Darcy's harsh criticism of women can be seen as his way of being openly direct about his opinions rather than "performing to strangers" by being guarded.
An example of Elizabeth's directness can be seen within this same moment at Netherfield. At one point when Miss Bingley is playing a lively Scottish song on the piano, Darcy asks Elizabeth if the music makes her want to "dance a reel," meaning jig (Ch. 10). Elizabeth interprets the comment as a means for Darcy to insult her taste and openly says so, thereby, in her mind, diverting any opportunity for Darcy to criticize and insult her. Others in the same situation as Elizabeth would not have been so bold as to openly accuse someone they had just met of willingly being insulting, showing us that Elizabeth also does not "perform to strangers."