Wordpress - How to change the markup Wordpress inserts for post images

As far as I know you could hook into the filter image_send_to_editor like this:

function html5_insert_image($html, $id, $caption, $title, $align, $url) {
  $url = wp_get_attachment_url($id);
  $html5 = "<figure id='post-$id media-$id' class='align-$align'>";
  $html5 .= "<img src='$url' alt='$title' />";
  $html5 .= "</figure>";
  return $html5;
}
add_filter( 'image_send_to_editor', 'html5_insert_image', 10, 9 );

For additional tags like data-pil-thumb-url and data-full-width and data-full-heightyou could add the appropriate code inside that function and add them to the $html5 string.

See also this page for an example featuring a caption <figcaption> at css-tricks or check this more detailed 'walk through'.

There's a filter called image_send_to_editor that lets you specify the markup. You will also need wp_get_attachment_metadata to retrieve width and height. It is called like this (untested):

   add_filter( 'image_send_to_editor', 'wpse_231693_img_markup', 10, 7 );
   function wpse_231693_img_markup ($html, $id, $alt, $title, $align, $url, $size ) {
       $metadata = wp_get_attachment_metadata ($id);
       $url = wp_get_attachment_url($id);
       $html = '<figure class="pil"><img src="' . $url . 
               '" data-pil-thumb-url="XXX" data-full-width="' . $metadata['width'] .
               '" data-full-height="' . $metadata['height'] .
               '" alt="' . $alt . '"></figure>';
       return $html;
   }

You will need some clever regex to construct XXX from $url, but I'll leave that to you.