What are the light conditions? Is it in a window or under a fluorescent light? What is your watering routine?

Privet--Chinese or otherwise are northern hardy evergreen shrubs which may be deciduous in some cases. This means that they aren't (sorry) houseplants. They need a period during the winter months to go dormant and keeping one inside won't do that.
One reason that it's probably dying from the top down is lack of high enough light and, additionally not enough water. In a normal indoor invironment, for something like that -- which is wrong anyway -- it would be a real water hog.

Bob