Получить значения из JSON страницы

Используйте json_decode с параметром true для ассоциативного массива

    $json = '{"ok":true,"result":[{"update_id":135028948,
"message":{"message_id":11,"from":{"id":603660417,"is_bot":false,"first_name":"\u0413\u0443\u0441\u044c","username":"Gusarich","language_code":"ru"},"chat":{"id":603660417,"first_name":"\u0413\u0443\u0441\u044c","username":"Gusarich","type":"private"},"date":1529566837,"text":"\u042f \u043d\u0443\u0431\u043b"}}]}';
    $array = json_decode($json,true);
    foreach ($array['result'] as $result){
        //do some here
        echo $result['update_id'];
        echo $result['message']['from']['id'];
        echo $result['message']['date'];
    }