Need help cutting out background from head shot

Wow, not to be over critical, but you have gone much too far retouching this ladies face. The idea with portraits processed through Photoshop is that they should not look as if they have been 'Photoshopped' . Of course your client may have asked to be made to look 25 years younger, but to be honest you have made her skin so perfect she resembles a waxwork figure not a real person anymore. I would suggest you keep some of the details that give this woman's face character like the frown line on her forehead and just tone down the crows feet around her eyes, also older people invariable have some signs of aging in the skin on the neck and this lady rivals a seventeen year old after your retouch. Sorry I'm not trying to be nasty, we all go over the top sometimes and I think it helps one improve when a slightly critical stance is taken. As for the hair look up how to manually draw in hair with a custom made brush, this sounds difficult but really isn't and you'd be amazed how much it is possible to improve even the look of the a difficult image like this one.

While making a duplicate image. changing its mode to CMYK and using Calculations with the settings shown here could be the basis for a mask in the RGB version that may rival your current results, I believe that the real elephant in the room is the hair color, made more distracting by the white background. A separate mask of the hair and change in hair color, once you know what it is, is vital. In fact, the same hair correction of your current image may be sufficient.

Wait a minute!

I just downloaded your raw file and, unmodified, brought it into PS. It looked like this:

I suggest you begin again.

Usually if I'm asked to change colors of elements in a photo,  I use adjustment layers. But that's just me. 

Where the image how can we even give you advice when we know not what your are dealing with.