Henry V was King of France in the mid-1840s, a time of growing tensions between the United States of Mexico and the Confederation of North America. Although Henry maintained the traditional friendship between France and the U.S.M., he was unable to help the Mexicans during the Rocky Mountain War due to the threat posed by the Germanic Confederation. (Although Sobel doesn't mention it, Henry was presumably also deterred by the likelihood that Great Britain would declare war on any European nation that aided Mexico against the C.N.A.)
Sobel does not mention Henry's family relationship to any other French kings. His name, Henry, suggests that he was a younger son who succeeded a deceased older brother as Dauphin of France; in that case, he may have been a younger son of Louis XVIII.
By the end of the Rocky Mountain War in 1855, Henry had been succeeded as King of France by Louis XIX.