The Keene Public Library has vital records on line for Keene; births 1753-1878, deaths 1742-1881 & 1887-1941. There are gaps here and you might find more information in the city and town eports. The City of Keene stopped including the vital records in their reports about 1941. the towns on the other hand are still providing vital records in their town reports.
The Historical Society of Cheshire County (HSCC) has an excellent compilation of all Cheshire County Vital Records that are in the town reports right uo to 2003.