Hi Zachary, I found a well-sourced family tree on this family on Ancestry and asked the author to contact you.
In 1860 Charles, wife and son were living in Country East of Township 6, Sierra Co., CA. Charles Burns, b. Ireland, 29, miner. Julia, wife, b. Ireland, 25. William, b. CA, 2.
In 1880 their family was living at Cox Bar and Snake Bar, Sierra Co., CA. Charles Burnes, b. Ireland, 50, miner. Julia, wife, b. Ireland, 42. William, b. CA, 22, teamster. Charles, b. CA, 20, mining. Mary, b. CA, 17. Julia, b. CA, 14. Lizzie [Elizabeth] b. CA, 13. Emma, b. CA, 10.
On 8 Oct 1892 a Charles Burnes, age=66, died in Sierra Co., born=Ireland. Source: Rootsweb, Sierra Co Deaths, Family History Library film #1293670.
Charles and Julia were born in Ireland. It might tell you where if you ordered a copy of Charles's death certificate. I'd contact the Sierra County Clerk, 530-289-3295.
OR California State Vital Records Office Vital Records, 916-445-1719.
I found more information for the following children; Charles and William. I'll share this with you if you're interested. Successful Searching!