Short answers, pulled from the story.
Jesus Christ walked the streets of Jerusalem during his ministry. The gospels record that he was brought to this city as a child to be presented at the Temple. He returned to attend the festival of Passover there.
James the Just led the early church in Jerusalem according to Pauline epistles. Origen wrote On the First Principles around 230 CE and argued that Jerusalem exists as a heavenly entity with transcendental features. This treatise suggested Israel belongs to the sphere of souls.
Emperor Constantine I and his mother Helena transformed Jerusalem into a pilgrimage center and endowed the city with churches and shrines around 325 CE. Helena searched for the True Cross with bishop Macarius of Jerusalem and removed a temple to Venus built over the site. An original church rose in 336 under Constantine's orders.
On the 15th of July 1099 the army captured Jerusalem during the First Crusade. Most city population died except Eastern Christians who fled in terror. Latin rulers exiled these remaining Eastern Christians from the city.
Five Christian communities hold rights within Holy Sepulchre now including Greek Patriarchate, Latins, Armenians, Copts, and Syriac Orthodox. These groups share access based on agreements upheld by British Mandate and Jordan historically. Franciscans serve as traditional Catholic custodians of holy places.