The below code will do so
public void DeleteFromBlob(string filename, string connectionString, string containerName)
{
// Retrieve storage account from connection string.
CloudStorageAccount storageAccount = CloudStorageAccount.Parse(connectionString);
// Create the blob client.
CloudBlobClient blobClient = storageAccount.CreateCloudBlobClient();
// Retrieve reference to a previously created container.
CloudBlobContainer container = blobClient.GetContainerReference(containerName);
// Retrieve reference to a blob named "myblob.csv".
CloudBlockBlob blockBlob = container.GetBlockBlobReference(filename);
// Delete the blob.
blockBlob.Delete();
}