Add image to post from external URL

I’m doing a plugin that can retrieve media from external APIs, and when I click on it (on the admin), it insert a new post with the media, its caption, etc.

So, now, I’m inserting a new post on click : works fine. I did something like that :

$title = $_POST['title'];
$content = $_POST['content'];
$id = $_POST['id'];
$imgSrc = $_POST['imgSrc'];

$my_post = array(
  'post_name'     => $id,
  'post_title'    => $title,
  'post_content'  => $content,
  'post_status'   => 'publish',
  'post_author'   => 1,
  'post_category' => array(8,39)
);

wp_insert_post( $my_post );

Now, that’s where I’m stuck. I have an image URL ($imgSrc), that I want to add to my post. What I want to do is :

1. upload this url in the media library : don’t know how to do that
2. attach it to the post : I saw something with wp_insert_attachment (but I don’t know how to handle it)

As I’m pretty noob to PHP, how can I do it ?

For those who would know :

$upload_dir = wp_upload_dir();
$image_data = file_get_contents($image_url);
$filename = basename($image_url);
if(wp_mkdir_p($upload_dir['path']))
    $file = $upload_dir['path'] . "https://wordpress.stackexchange.com/" . $filename;
else
    $file = $upload_dir['basedir'] . "https://wordpress.stackexchange.com/" . $filename;
file_put_contents($file, $image_data);

$wp_filetype = wp_check_filetype($filename, null );
$attachment = array(
    'post_mime_type' => $wp_filetype['type'],
    'post_title' => sanitize_file_name($filename),
    'post_content' => '',
    'post_status' => 'inherit'
);
$attach_id = wp_insert_attachment( $attachment, $file, $post_ID );
require_once(ABSPATH . 'wp-admin/includes/image.php');
$attach_data = wp_generate_attachment_metadata( $attach_id, $file );
wp_update_attachment_metadata( $attach_id, $attach_data );

set_post_thumbnail( $post_ID, $attach_id );