How CSP find the private key of certificate to perform cryptographic operations?

When you import a certificate to the system store, Windows creates a BLOB structure that contains an encoded certificate itself and it properties. BLOB has following structure:

property1_id (4 bytes)
reserved = 0x00000001
property1_length (4 bytes)
property1_data[property1_length]
...
cert_property_id = 0x00000020
reserved = 0x00000001
cert_data_length (4 bytes)
cert_data[cert_data_length]

Therefore, if you want that your imported certificate to have link to a private key, you need to set CERT_KEY_PROV_INFO_PROP_ID. You can achieve that with CRYPT_KEY_PROV_INFO structure and CertSetCertificateContextProperty function.

For example:

#include <Windows.h>
#include <wincrypt.h>

void SetKeyLink()
{
    HCERTSTORE hStore = NULL;
    CRYPT_KEY_PROV_INFO key_prov_info = { 0 };
    PCCERT_CONTEXT pCertContext = nullptr;
    std::vector<BYTE> der_encoded_cert;

    hStore = CertOpenSystemStore(NULL, L"MY");
    if (!hStore)
    {
        goto Exit;
    }

    der_encoded_cert = LoadFromFile();

    pCertContext = CertCreateCertificateContext(X509_ASN_ENCODING, der_encoded_cert.data(), der_encoded_cert.size());
    if (!pCertContext)
    {
        goto Exit;
    }

    /* For legacy CSP */
    key_prov_info.dwProvType = PROV_RSA_AES; // Or YOUR_PROVIDER_TYPE
    key_prov_info.dwKeySpec = AT_SIGNATURE; // Or AT_KEYEXCHANGE
    key_prov_info.pwszContainerName = L"Your_key_name";
    key_prov_info.dwFlags = CERT_SET_KEY_PROV_HANDLE_PROP_ID;
    key_prov_info.cProvParam = 0;
    key_prov_info.pwszProvName = nullptr;
    key_prov_info.rgProvParam = 0;

    /*
    Or if you use CNG Key storage provider:

    // Or L"Your_CNG_key_storage_provider_name"
    key_prov_info.pwszProvName = L"Microsoft Software Key Storage Provider"; 
    key_prov_info.pwszContainerName = L"Your_key_name";
    key_prov_info.dwFlags = CERT_SET_KEY_PROV_HANDLE_PROP_ID;
    key_prov_info.dwProvType = 0;
    key_prov_info.dwKeySpec = 0;
    key_prov_info.cProvParam = 0;
    key_prov_info.rgProvParam = 0;
    */

    if (!CertSetCertificateContextProperty(pCertContext, CERT_KEY_PROV_INFO_PROP_ID, 0, &key_prov_info))
    {
        goto Exit;
    }

    if (!CertAddCertificateContextToStore(hStore, pCertContext, CERT_STORE_ADD_ALWAYS, NULL))
    {
        goto Exit;
    }

    std::cout << "success";

Exit:

    if (pCertContext)
    {
        CertFreeCertificateContext(pCertContext);
    }

    if (hStore)
    {
        CertCloseStore(hStore, 0);
    }
    return;
}

For now your certificate will look something like this (sorry for not English):

When Windows wants to get private key, it calls CryptAcquireCertificatePrivateKey which in turn calls CertGetCertificateContextProperty(..., CERT_KEY_PROV_INFO_PROP_ID, ...).