The haunted psychologist putting himself in the mind of a psychotic murderer by analysing a bloody crime scene and coming to amazingly precise conclusions about the killer's character and motive has become a movie trope frequently seen in dramas such as Cracker, Silence of the Lambs and the recent Netflix series Mindhunter. The latter dramatises the work of real life FBI agents John Douglas and Robert Ressler who in the late 1970's set out to interview the most notorious murderers in the country. They were looking for patterns that connected a killer’s life and personality with the nature of his crimes. After they had interviewed [just] 36 murderers they were convinced they'd found patterns linking the actions and character of what they called a spree murderer - what we now call a serial killer.
Their conclusions and methodology formed the basis of offender and criminal profiling which over the years has been widely drawn upon by police forces. In most cases, a local force will send details of the crime (photos, reports, forensic data etc) to a profiler who makes predictions about the personality, behavioral, and demographic characteristics of the perpetrator. These predictions are then reported to investigating officers who can use this information to narrow down the list of subjects or help them find new avenues of investigation. There is no doubt that police forces found this information useful, one 1995 study found that found that 83% of a sample of 184 police officers in the UK claimed that profiling was operationally useful and 92% reported that they would seek CP advice again.
Unfortunately, there is much to be skeptical of in profiling. Most skeptics will be familiar with cold-reading (used by psychics to give the illusion they have secret knowledge when they are simply extrapolating from everyday clues we give away in dress, speech, mannerisms etc) and profiling has been compared to that. Too often police forces fall into same traps that people who believe in psychics do - counting the hits and forgetting the misses; interpreting conclusions about the killer, that are really just Barnum statements, in hindsight, and accepting vague and meaningless conclusions as being precise and authoritative. Several reports from psychologists around the world have called for caution in the use of profiling in detection. For instance, Snook et al (2008) stated that that '[Criminal Profiling] should not be used as an investigative tool because it lacks scientific support'. For a good summary of the field, read Malcolm Gladwell's article in the 2007 New Yorker: https://www.newyorker.com/magazine/2007/11/12/dangerous-minds.