what do you mean by private bitcoin core chain?
1) bitcoin core it's just a client, so use bitcoin blockchain term.
2) do you want to fork bitcoin codebase? In this case you have to change more parameters than genesis block: port number, magic message number ect.
3) But if you just want to run private bitcoin blockchain for your own testing or something similar, you don't have to change anything just start daemon with -regtest option.

    const char* pszTimestamp = "The Times 03/Jan/2009 Chancellor on brink of second bailout for banks";
    const CScript genesisOutputScript = CScript() << ParseHex("04678afdb0fe5548271967f1a67130b7105cd6a828e03909a67962e0ea1f61deb649f6bc3f4cef3 8c4f35504e51ec112de5c384df7ba0b8d578a4c702b6bf11d5f") << OP_CHECKSIG;