How do biochemists seperate nucleotides from DNA to add to the PCR
You just buy the nucleoside triphosphates from your favorite supplier, they're commercially available.

As far as I know the NTPs you can buy are still produced biologically and not synthesized (see this example from Sigma Aldrich, which states a microbial source). So you grow a large amount of bacteria, lyse the cells to get at the contents and then digest the DNA or RNA with an appropriate enzyme that cuts it into single nucleotides. To get the triphosphate from that you have to phosphorylate the monophosphates again with the appropriate kinase.