There are 10 US Post Offices in Johnson County serving a total of 146,641 residents equating to 14,664 residents per post office. It's estimated that approximately 234,055 packages are handled across the 10 county post offices annually. Johnson County is a county located in the U.S. state of Texas. As of the 2010 census, its population was 150,934. Its county seat is Cleburne. Johnson County is named for Middleton Johnson, a Texas Ranger, soldier, and politician. Johnson County is included in the Dallas-Fort Worth-Arlington, TX Metropolitan Statistical Area.