A much cheaper and probably better option than a private guide is the audio guide.
For 4 euros you can hire a small audio device that plays commentaries as you go around.

There are about 40 numbered points at the Alhambra and when you press the number on your MP3 player device you can hear a commentary.
The production quality is high and it is just as good if not better than a real human.
You can get them from the information desk next to the ticket windows.
You have to leave your passport as security.

If you are a real cheapskate you can eavesdrop on other tour guides.
I do this occasionaly, some of the guides are good and some are very bad.

If you really do want a private tour with a human guide here is some info and a telefone number at the end.

Transport is not included.
Private Daytime Tour Timing: 3 hrs approximately.
Private Night Tour Timing: 1.30 hrs approximately.

Private Daytime Tour Itinerary: Nasrid Palaces, Charles V Palace, Medina and Generalife Gardens and Palace.

Private Night Tour Itinerary: In the Alhambra, Only Nasrid Palaces and Charles V Palace.
Group: from 1 to 30 persons (max. 30 persons) per guide.

Language: English, French and Spanish guaranteed.
Ask for German, Portuguese, Danish, Italian, Arabic, Japanish, Swedish, Dutch or Russian guided tours.

Meeting Point: In the Alhambra and Generalife Access Pavilion, near the Tourism Information Office.

Contact: Tel: 0034 958225245/26.

Mobile: 00 34 677417790

There is an excellent Virtual Walking tour of the Alhambra. Click here