He was educated in Milwaukee Public Schools, and in the spring of 1870 joined his father in running the family brick factory, gradually taking management of it over along with his younger brother John Burnham.
When elected to the Assembly in 1877 to represent the 8th Milwaukee County district (the Eighth and Eleventh Wards of the City of Milwaukee), Burnham ran as an "Independent Greenback", winning with 771 votes to 537 for the incumbent, Democrat Peter Salentine and 87 for Socialist William Stange.
Burnhanm was generally identified as a Republican; he served for some time on the Milwaukee Board of School Directors, as had his father before him.
[3] Their father's estate, when it was filed for probate in 1889, was estimated to be worth US$2 million, of which Charles was one-fourth heir (he was also an executor).
Burnham married Virginia Johnston of Mackinac Island, Michigan; they had three children, and were members of the local Congregationalist church.