I'm trying to figure out if Kirk had a 5 year mission all to himself. In the original series, was the entire 5 year mission of the Enterprise and her crew commanded by Kirk, or did he inherit the remainder of Pike's mission? We know Kirk placed others in command several times during the series, but did Kirk have a 5 year mission from beginning to end?
Yes, Kirk did command the Enterprise through her entire five year mission and for many years after that. In 2265, at 32 years of age, he assumed command of the Constitution-class starship USS Enterprise (NCC-1701), after Christopher Pike. (TOS: "The Menagerie, Part I")
Memory Alpha's mentions this in Kirk's biography on the Enterprise's Five Year mission, from 2265 to 2270.
Kirk famously commanded the Enterprise and her namesake over the course of three decades, but it was her historic five-year mission from 2265 to 2270 that made him a legend in space exploration. In addition to his overall mission statement, "to explore strange new worlds, to seek out new life, etc.", Kirk received standing orders to investigate all quasars, and quasar-like phenomena.
Kirk's living quarters aboard the Enterprise were on Deck 5, room "3F 121". (TOS: "The Galileo Seven", "Journey to Babel"; VOY: "Q2") During this historic mission, Kirk violated the Prime Directive on at least one occasion in order to ensure the survival of the Pelosian race. Years prior to this, Kirk also prevented the extinction of the Baezians and the Chenari. (VOY: "Q2")