Educated in their community's public schools, Henry and his siblings relocated with their parents to Pittsburgh sometime around 1830, where his father launched a new business venture, the opening of the Mansion House at Fifth Avenue and Wood Street (later the site of the First National Bank).

His father, who was politically active throughout Henry Weaver's youth, was a member of the Whig Party who was elected as sheriff of Allegheny County, Pennsylvania in 1840.

[1][2][3] At the age of eighteen, Henry and his brother began their own careers by opening a canal and river supply business on Tenth Street.

[1][2] In 1856, he served as a delegate to the first Republican National Convention, during which John C. Frémont was chosen as the party's candidate for the upcoming election for the office of president of the United States.

[1][2] In 1860, President Abraham Lincoln appointed him as an assessor of internal revenue for Western Pennsylvania, a post he held until 1869.