make blog private while under construction

There is a handful of plugins that are designed to help out with that while you build the site. http://wordpress.org/plugins/search.php?q=under+construction

You might also take a look at plugins of the maintenance mode variety.
http://wordpress.org/plugins/search.php?q=maintenance+mode&sort=

Some of those are quite creative. A couple of random examples of what you might find:

WP Maintenance Mode

WP Construction Mode

You’re welcome!