Most of the area we know as Arizona became part of the United States via the Mexican War in 1848. The rest - from the Gila river south to the Mexican border, which includes what is now Cochise County - was obtained in the Gadsden Purchase in 1853. In 1863, Arizona was made a territory, and became the 48th state of the union in 1912.
Cochise County was originally part of Pima County, but was split off in 1881, and Tombstone was designated as the county seat. By 1929, between the silver mines petering out and Prohibition, Tombstone was in a serious decline and the county seat was moved to Bisbee, where it remains today.
1881 Historical Note
In February 1881, the newly-formed Cochise County needed a sheriff, and Wyatt Earp and John Behan both wanted the job. However, Behan had political connections which "allowed" Gov. Fremont to appoint Behan as sheriff, bypassing the normal election process. This, and their mutual interest in actress Josephine Sarah Marcus (whom Wyatt later married), caused Behan and Wyatt to became serious enemies.
Behan wasn't very interested in enforcing the law at all, but a lot of Tombstone residents were. This caused the town to split into two factions - one aligning with the Earps (Virgil was now the city marshal) and law enforcement, and the other with the lawless group of the Clantons and their "Cowboys" (cattle rustlers), Behan, and a host of others who wanted Cochise County, especially Tombstone, to remain fundamentally lawless.
This friction culminated at the Gunfight at the OK Corral on Oct 26, 1881. It was a busy year.