Still itching to strike it rich, Billy and his stepfather finally decided to go to Arizona--to the recently discovered silver deposits in a place that was called "Globe City." They arrived there about 1876, and it was in Globe that Billy actually ran afoul of the law for a serious offense: stealing horses. Apparently, he also rustled cattle. He was rowdy and often involved in gambling and similar nefarious pursuits.
In August 1877 Billy was at Fort Grant (new Fort Grant, established by General Crook south of Mt. Graham in 1872). The Kid was tormented in a bar there by another rowdy, by the name of Frank Cahill. The Kid did not take kindly to the razzing, and so he shot and killed Cahill. The Kid was captured and jailed, but he escaped and went to Lincoln County, New Mexico.
For awhile the Kid rambled around the Ruidoso Valley and was then employed by John H. Tunstall. It was Tunstall that was responsible to a great extent for the famous "Lincoln County War" in which the Kid played a principal part. There were a lot of violent incidents in which the Kid was involved and, finally, he was captured. He escaped this capture also, but was eventually shot down by Sheriff Pat Garrett on 14 July 1881 in Fort Sumner, New Mexico.
For more about this pathetic character you might want to read:
Mullin, Robert N., ed. Maurice G. Fulton's History of the Lincoln County War. Tucson: Univ. of Arizona Press, 1968.
Weddle, Jerry. Antrim Is My Stepfather's Name: The Boyhood of Billy the Kid. Foreword by Robert M. Utley. Tucson: Arizona Historical Society, Historical Monograph No. 9, 1993.
A web page about a personal reminiscence regarding the Billy the Kid.