The "Autobiography of James T. Kirk", told in-universe by Kirk himself and licensed by Paramount has some additional information to offer about Kirk's ethnicity. In short, his primary ethnic makeup appears to be broadly Caucasian (presumably Scottish given that the Kirk name evidently predates his American ancestry in the 1800s) as well as part-Italian(?) and part-Sioux on his mother's side.
The house was four bedrooms, two floors, brick and wood. It was built
using authentic materials and was a perfect copy of the house that had
stood on the property for over 100 years in the 19th and 20th
centuries. The property had belonged to seven generations of Kirks; it
was family legend that my
great-great-great-great-great-great-great-grandfather, Franklin Kirk,
purchased the farm in 1843 from Isaac Cody, who was the father of
William F. “Buffalo Bill” Cody. My ancestors in the modern era let
caretakers manage it, until my grandparents moved back there when they
retired. My father, George Kirk, also always had a strong desire to
live there.
He had grown up as one of the original “Starfleet brats”; his father,
Tiberius Kirk, was already in his twenties when Starfleet Academy was
founded, and though he applied, he wasn’t accepted. Still wanting to
get out into space, Tiberius signed on in ordnance and supply,
eventually serving on several of the then-new starbases. He met and
married my paternal grandmother, Brunhilde Ann Milano, a nurse, on
Starbase 8. My father was born there on December 13, 2206.
and
My mother, born Winona Davis, was also from a spacegoing family; her
father, James Ogaleesha Davis (his middle name, as befit his heritage,
was Native American Sioux, although I never did find out what it
meant), was in the first graduating class of Starfleet Academy; his
wife, Wendy Felson, was in the third. My maternal grandfather was an
engineer, my maternal grandmother a physician, and their daughter, my
mother, attended the academy and decided she wanted to be an
astrobiologist. She was four years younger than my father, and had him
as an instructor in her Introduction to Federation History class.