This Kohana forum post indicates that it should be possible. Given the similarity to your code, I'm guessing you already found that. As that's not working for you, you could try cURL:
$postData = array('myfile' => '@../../testimage.jpg');
$uri = 'files';
$ch = curl_init();
curl_setopt($ch, CURLOPT_URL, $uri);
curl_setopt($ch, CURLOPT_POSTFIELDS, $postData);
curl_setopt($ch, CURLOPT_RETURNTRANSFER, true);
curl_setopt($ch, CURLOPT_FOLLOWLOCATION, true);
curl_setopt($ch, CURLOPT_POST, true);
$response = curl_exec($ch);
If you want to use the Kohana Request, you could try building your own multipart body using this code (I don't have the proper setup to test it right now, but it should be close to what you need):
$boundary = '---------------------' . substr(md5(rand(0,32000)), 0, 10);
$contentType = 'multipart/form-data; boundary=' . $boundary;
$eol = "\r\n";
$contents = file_get_contents('../../testimage.jpg');
$bodyData = '--' . $boundary . $eol;
$bodyData .= 'Content-Type: image/jpeg' . $eol;
$bodyData .= 'Content-Disposition: form-data; name="myfile"; filename="testimage.jpg"' . $eol;
$bodyData .= 'Content-Transfer-Encoding: binary' . $eol;
$bodyData .= $contents . $eol;
$bodyData .= '--' . $boundary . '--' . $eol . $eol;
$request = new Request('files');
$request->method(HTTP_Request::POST);
$request->headers(array('Content-Type' => $contentType));
$request->body($data);
$request->execute();