Most records of Clay County are available at the Archives or in the Clerk's Vault at the Courthouse.
However, there are other public records that you may be interested in:
Duval County records
Clay County wasn't formed until 1859. Duval was the governing county prior to that, and as such, early records are stored in the Clerk's vault there. It is somewhat tedious to determine which records are related to citizens who lived in the Clay area, since there are generally no clues given.
In addition, some very early history books make reference to a box of records related to Clay County in the Duval Clerk's vault. If you should ever run across that box, the Clay County Archivist would be most appreciative of a call. The whereabouts of the box has not been known for quite some time.
In any case, the Archives has:
* Indexes to Duval Wills Vol C
* Index to Duval Probate Book A (photographs taken from microofilm)
* Complete transcription of Duval Marriages through 1860 (in searchable PDF)
St. Johns County records
Duval County was formed in 1822. Prior to that, the governing county was St. Johns. The St. Augustine Historical Society may be able to help you with records from that era.
State Archives of Florida
The State Archives of Florida is custodian for a vast collection of early records.
In particular, death records were completed at the state level and not at the county. Volunteers have transcribed Clay County Death Records from the state death records.