I've followed the guide here

http://www.windowsazure.com/en-us/develop/net/how-to-guides/queue-service/

Here is my storage connection string "DefaultEndpointsProtocol=https;AccountName=***;AccountKey=***"

And here is my code that i have successfully debugged locally and tested it works and also deployed to Azure.

        int Queue()
        {
            CloudStorageAccount storageAccount = CloudStorageAccount.Parse(RoleEnvironment.GetConfigurationSettingValue("StorageConnectionString"));
            CloudQueueClient queueClient = storageAccount.CreateCloudQueueClient();
            CloudQueue queue = queueClient.GetQueueReference("myqueue");
            queue.CreateIfNotExist();

            CloudQueueMessage message = new CloudQueueMessage("Hello, World");
            queue.AddMessage(message);

            CloudQueueMessage retrievedMessage = queue.GetMessage(TimeSpan.FromSeconds(1));
            Thread.Sleep(2);
            var msgs = queue.GetMessages(1, TimeSpan.FromSeconds(60));
            msgs.ToList().ForEach(msg => queue.DeleteMessage(msg));
            queue.Delete();
            return 0;
        }