'Doing gender' is a phrase used to frame gender in the context of a cluster of actions that individuals undertake, as opposed to a biologically based category. In this model gender is not primarily something that emerges naturally from our biology nor a simple set of intuitive rules; as individuals grow up and learn how to interact socially they are also, slowly and sometimes painfully, learning how to act the role culturally assigned to their gender.
Those who try to act outside of those roles will be corrected by parenting, peer pressure, direction from authority figures, school and workplace rules, laws, alienation, microaggression, harassment, and abuse. Most people do not spend much time trying.
The term doing gender was coined in an article of the same title published in 1987, which has since become one of the most cited articles in the field of gender studies (it's also well known and oft referenced in the field of sociology). Because of its popularity and widespread application it has covered a lot of different ground as it has been reinterpreted by different writers. You may find the same general idea referred to as gender performativity or the social construction of gender.
Reference: West, Candace; Zimmerman, Don H. (1987). Doing Gender. Gender & Society 1 (2): 125–151.
Brevity Quest 2016