Help with verifying google recaptcha in a custom form

I am trying to add a recaptcha to my custom field. I have created a custom form in a page template and this the part of my code I think is relevant (don't worry that API key is MerchantOne's public demo api so it's not personal):

// API Setup parameters
$gatewayURL = 'https://secure.merchantonegateway.com/api/v2/three-step';
$APIKey = '2F822Rw39fx762MaV7Yy86jXGTC7sCDy';


// If there is no POST data or a token-id, print the initial shopping cart form to get ready for Step One.
if (empty($_POST['DO_STEP_1'])  empty($_GET['token-id'])) {

    print '  !DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"';
    print '
    html
      head
        meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /
        titleHansen Hunter Collect non-sensitive Customer Info/title
      /head
      body
      pWe are pleased to offer the option for you to securely pay your Hansen Hunter  Co. invoice online./p
      h2 class="pad"Invoice Details/h2


        form action="" method="post"
          table
          trtdinput type="text" name="shipping-address-first-name" placeholder="First Name" required/td/tr
          trtdinput type="text" name="shipping-address-last-name" placeholder="Last Name" required/td/tr
          trtdinput type="text" name="shipping-address-company" placeholder="Company" required/td/tr
          trtdinput type="text" name="billing-address-phone" placeholder="Phone Number" required/td/tr
          trtdinput type="text" name="billing-address-email" placeholder="Email Address" required/td/tr
          trtdinput type="text" name="billing-invoice" placeholder="Invoice or Customer Number"/td/tr
          trtdinput type="text" name="billing-amount" placeholder="Total Amount" required/td/tr

          trtdh4br /Payment Information/h4/td/tr

          trtdinput type="text" name="billing-address-first-name" placeholder="Cardholder First Name" required/td/tr
          trtdinput type="text" name="billing-address-last-name" placeholder="Cardholder Last Name" required/td/tr
          trtdinput type="text" name="billing-address-address1" placeholder="Address" required/td/tr
          trtdinput type="text" name="billing-address-address2" placeholder="Address 2"/td/tr
          trtdinput type="text" name="billing-address-city" placeholder="City" required/td/tr
          trtdinput type="text" name="billing-address-state" placeholder="State" required/td/tr
          trtdinput type="text" name="billing-address-zip" placeholder="Zip/Postal Code" required/td/tr
          trtdinput type="text" name="billing-address-country" placeholder="Country" value="US" required/td/tr
       trtddiv class="g-recaptcha" data-sitekey="6Lc2MiUTAAAAAGJeGAlku50gFi7hFKvIitP_-3e0"/div/td/tr
          trtdinput type="submit" class="btn pad" value="Continue"input type="hidden" name ="DO_STEP_1" value="true"/td/tr
          /table
        /form

    ';
}else if (!empty($_POST['DO_STEP_1'])) {

    // Initiate Step One: Now that we've collected the non-sensitive payment information, we can combine other order information and build the XML format.

I've added this code to my functions.php to add the google api:

add_action('wp_head','hook_recaptca');

function hook_recaptca() {

    $output="script src='https://www.google.com/recaptcha/api.js'/script";

    echo $output;
} 

This allows my recaptcha to show up. The problem is it's not verifying yet.

I know I have these 2 pieces of code to still add:

/**
 * Send a GET request to verify CAPTCHA challenge
 *
 * @return bool
 */
function captcha_verification() {

    $response = isset( $_POST['g-recaptcha-response'] ) ? esc_attr( $_POST['g-recaptcha-response'] ) : '';

    $remote_ip = $_SERVER["REMOTE_ADDR"];

    // make a GET request to the Google reCAPTCHA Server
    $request = wp_remote_get(
        'https://www.google.com/recaptcha/api/siteverify?secret=your_secretresponse=' . $response . 'remoteip=' . $remote_ip
    );

    // get the request response body
    $response_body = wp_remote_retrieve_body( $request );

    $result = json_decode( $response_body, true );

    return $result['success'];
}

and

/**
 * Verify the CAPTCHA answer
 *
 * @param $user string login username
 * @param $password string login password
 *
 * @return WP_Error|WP_user
 */
function validate_captcha( $user, $password ) {

    if ( isset( $_POST['g-recaptcha-response'] )  !captcha_verification() ) {
        return new WP_Error( 'empty_captcha', 'strongERROR/strong: Please retry CAPTCHA' );
    }

    return $user;
}

but I'm not sure how to attach them (what hook to use?) in my form so that a user is stopped if they don't fill the checkbox. I've see tutorials and questions about adding to the login form or the registration form or the comments, but can't seem to figure out how to do it to my custom form.

In case it helps, this the entire php that merchant one supplied for their integration is here:

http://pastebin.com/mR5jf5B3