Looking at the comments above, it is possible that the $url you are using is not valid, check it out by pointing your browser there and see what happens.
If you are sure that the $url is fine, but you still get the 404 Not Found error - verify that you have proper Laravel routing defined for that address. If the routes are fine, maybe you forgot to do
composer dump-autoload
after making modifications in your routes.php. If so, try the above and refresh the browser to see if it helps.
Furthermore, bear in mind that using your current function, you can submit only GET requests. What is more, this function might not be available for fetching remote urls, on some hosting servers due to security reasons. If you still want to use it, it'd be good to check
if($json !== FALSE)
before you process the $json response. If the file_get_contents fails it will return false.
Reffering to the part of your question
what is the proper way for it to be flexible and be able to get the JSON string with any $url
I'd suggest using cURL, as a standard and convenient way to fetch remote content. Using cURL you have better control over the process of sending the http request and receiving the "answer" it returns. Personaly, in my Laravel 4 apps I often use this package jyggen/curl. You can read the docs for it here: jyggen docs
If you are not satisfied with cURL and you want greater control try Guzzle As the authors state, Guzzle is a PHP HTTP client & framework for building RESTful web service clients.