If you have a vertical mill then gravity will have an effect on the Z axis. You may not need backlash compensation. If you have a spring set it so that the spindle is pushed down so that together with gravity it acts as a backlash eliminator. Having the spring opposing gravity as is usual in a drill will result in variable backlash so backlash compensation may need to be variable. The forces on the Z axis may in some positions push it down, in others they may balance and at the other end of travel they may push it up.
If you are using a knee for the Z axis movement then gravity is usually acting as a backlash eliminator so no problems with backlash are the most likely outcome.
Is it possible that the chuck movement of 0.005" is due to loose bearings? If so this should be sorted out as loose bearings are not good for milling in general.
Martin