Using your current code, i've made a small snippet. You might need to change a few things according to your needs, but for this matter it works as intended. To make a good use of the CURL request and not make and use repetitive code, I would put it in a function, in this case MakeRequest($endpoint, $data)

Where $endpoint is a String and $data should be an Array

In order to open and send a direct message to a user, you need these endpoints.

For creating a new direct message

POST /users/@me/channels

For sending messages:

POST /channels/{channel.id}/messages

<?php
function MakeRequest($endpoint, $data) {
    # Set endpoint
    $url = "https://discord.com/api/".$endpoint."";

    # Encode data, as Discord requires you to send json data.
    $data = json_encode($data);

    # Initialize new curl request
    $ch = curl_init();
    $f = fopen('request.txt', 'w');

    # Set headers, data etc..
    curl_setopt_array($ch, array(
        CURLOPT_URL            => $url, 
        CURLOPT_HTTPHEADER     => array(
            'Authorization: Bot token',
            "Content-Type: application/json",
            "Accept: application/json"
        ),
        CURLOPT_RETURNTRANSFER => 1,
        CURLOPT_FOLLOWLOCATION => 1,
        CURLOPT_VERBOSE        => 1,
        CURLOPT_SSL_VERIFYPEER => 0,
        CURLOPT_POSTFIELDS => $data
        CURLOPT_STDERR         => $f,
    ));

    $request = curl_exec($ch);
    curl_close($ch);
    return json_decode($request, true);
}

# Open the DM first
$newDM = MakeRequest('/users/@me/channels', array("recipient_id" => "ID From the user"));

# Check if DM is created, if yes, let's send a message to this channel.
if(isset($newDM["id"])) {
    $newMessage = MakeRequest("/channels/".$newDM["id"]."/messages", array("content" => "Hello World."));
}
?>

Heads up: Due security and privacy matters, a direct message might not open if:

• The user doesn't share the same server as your bot.
• The user has turned off DM's from server members.
• The user has blocked your bot.