The C API for processing JSON is Jansson. C applications like libblkmaker use cURL for making the calls and Jansson for interpreting the JSON that cURL fetches.

For example basic usage (which can be easily modified for Bitcoin RPC), see the Jansson example github_commits.c and the associated tutorial.

Here's an example of how to make JSON RPC calls with cURL, the output of which Jansson could process:

#include <stdlib.h>

#include <curl/curl.h>

int main()
{
    CURL *curl = curl_easy_init();
    struct curl_slist *headers = NULL;

    if (curl) {
    const char *data =
        "{\"jsonrpc\": \"1.0\", \"id\":\"curltest\", \"method\": \"getinfo\", \"params\": [] }";

    headers = curl_slist_append(headers, "content-type: text/plain;");
    curl_easy_setopt(curl, CURLOPT_HTTPHEADER, headers);

    curl_easy_setopt(curl, CURLOPT_URL, "http://127.0.0.1:8332/");

    curl_easy_setopt(curl, CURLOPT_POSTFIELDSIZE, (long) strlen(data));
    curl_easy_setopt(curl, CURLOPT_POSTFIELDS, data);

    curl_easy_setopt(curl, CURLOPT_USERPWD,
             "bitcoinrpcUSERNAME:bitcoinrpcPASSWORD");

    curl_easy_setopt(curl, CURLOPT_USE_SSL, CURLUSESSL_TRY);

    curl_easy_perform(curl);
    }
    return 0;
}

It will print, like the cURL example, something like this:

{"result":{"balance":0.000000000000000,"blocks":59952,"connections":48,"proxy":"","generate":false, "genproclimit":-1,"difficulty":16.61907875185736},"error":null,"id":"curltest"}