how do to get the carbon out of the ashes
Any more details on how to get the carbon out of the ashes? I am assuming that the normal cremation process in in an oxidizing atmosphere which would convert any organic carbon to CO(2). I would not expect any elemental carbon to remain in the ashes. If the temperature were hot enough to melt gold crowns (1063 C) , I would not even expect carbon in the form of mineral carbonates to remain, Calcium and Sodium carbonates decompose at 894 C and 400 C respectively.
Upon further investigation, I find that for wood ash:
"Carbonates are presumably formed at low temperatures in a quiescent atmosphere when the combustion products, primarily carbon dioxide, surround the wood grains. "
"Ash formed at high temperatures in an oxidizing atmosphere consist primarily of metal oxides."
"It appears that when the ash is left standing in air, calcium oxide reacts with atmospheric water vapor to form calcium hydroxide, however calcium ..."
My interpretation is that creamation at low temperatures, say , 500 C would leave carbon in the form of carbonate in the ashes, which originated from the corpse of the deceased. However, creamation at over the melting point of gold, would decompose the carbonates into oxides, leaving no carbonate, nor any recoverable form of carbon that I know of. Furthermore, ref  above says formation of hydroxides from the oxides takes place, presumably during storage. It is well know that hydroxides absorb atmospheric CO(2) carbon dioxide, reforming carbonates. This significance of this taking place is that the ash no longer contains carbon (as carbonates) originating from the deceased, but originating from atmospheric carbon dioxide. The diamond may or may not contain carbon from the deceased depending upon the temperature of cremation. Perhaps a mortician can answer that.
 M. Ahendra, et al, WOOD ASH COMPOSITION AS A FUNCTION OF FURNACE TEMPERATURE, pp13.
 ibid, pp11.