elijah d. cochran. died dec. 22, 1863 buried in camp chase, columbus ohio

He was my great-grandfather and I'm trying to find out info about him. He was in the infantry & the tombstone says rank of private. He may have been with the 5th Kentucky Mounted Infantry. How do I go about finding information on him? All our relatives are dead and I have no records.

Thank you for your help.
Linda Cross