Carter Co., Ky death records list Alexander Hamilton, age 30, born in Bath Co., Ky, farmer, married, son of William Easther Hamilton, resident of Carter Co., died in Carter Co. on 11 Feb 1853, cause of death typhoid fever.

(It was probably meant to say William and Esther Hamilton, since all of the other records mention two parents, but I’m transcribing what it said.)

Bath Co., Ky marriage records list the marriage on 21 March 1849 of Alexander Hamilton and Mary Cartmill.

1850 census of Bath Co., Ky, division 2, enumerated 24 July, lists on p. 16 as family #110:
Alexander Hamilton, 27, farmer; Mary J., 20; David W., 1. All born in Kentucky.