Short answers, pulled from the story.
Claire Temple is a nurse who finds Matt Murdock beaten and lying in a dumpster during the first season of Daredevil. She serves as an amalgam of two distinct Marvel Comics characters named Claire Temple and Night Nurse.
Rosario Dawson joined the cast of Daredevil in June 2014. Her role was revealed to be that of Temple in October of that year.
Claire Temple moves to Harlem because the hospital tries to cover up a coworker's death following an attack by The Hand. She quits her job at Metro-General Hospital and relocates where her mother runs a diner.
Marvel Television head Jeph Loeb explains that Dawson fits so well in those worlds that they wanted her to serve as a thread connecting them all. She appears in the second season of Daredevil, Luke Cage, Iron Fist, and the crossover miniseries The Defenders.
Claire Temple falls for Cage before he gets arrested but they argue and ask to take a break from each other by the end of the second season of Luke Cage. They have a fight in their final scene which ends their relationship completely.