What word would you use for a piece of writing which aims to describe a person? In particular, their personality. I do not think it is the same as 'biography', because a biography is an account of the events in a person's life, which may or may not be important for describing that person's character. I have been figuratively using the word 'portrait', or phrase 'written portrait', but I am keen to hear other suggestions.
A profile is a snapshot of a particular person that attempts to capture some characteristic about them.
Here are a couple of relevant entries from the Oxford English Dictionary under "profile, n.":
II. 10.a. A short biographical sketch or character study, esp. of a public figure
c. A record or report of a person's psychological or behavioural traits, personal preferences, etc., typically based on the analysis of his or her responses in a questionnaire or test. Cf. sense 15 and profiling n. 5.
d. Originally U.S. In a criminal investigation: a description of the probable psychological and behavioural characteristics of an unknown perpetrator, created to help identify suspects. Cf. offender profiling n. at offender n. Compounds.
So periodicals may publish profiles of various leaders that follow a specific person through a routine or through recent accomplishments. In another context, psychologists or criminologists may create a profile of either an actual or a hypothetical person of interest. The focus tends to be their character, or (figuratively) what makes them tick.
For example, EGMN (sourcing from the New York Times) defines a profile feature in terms of a target person's background and character:
A “profile feature” is a newspaper article that explores the background and character of a particular person (or group). The focus should be on a news angle or a single aspect of the subject’s personal or professional life. The article should begin with the reason the subject is newsworthy at this time, and should be based (not exclusively) on an extensive interview with the subject.
Publications often refer to these explicitly as profiles. The New Yorker has a Profiles section, and searching any given publication for a "profile" is likely to yield results like this top result from the Los Angeles Review of Books: "The Poetics of Trauma and Life After Rape: A Profile of Frances Driscoll."