Open Mozilla Firefox automatically in Private Mode

Click Menu -> Options

Click Privacy & Security
Under section History, Click Firefox will drop down list box
Select Never remember history

Prompt will popup asking to restart Mozilla Firefox Browser
Select Restart Firefox now

Now Mozilla Firefox Browser will start in normal mode with the settings of private browsing mode.

Start Mozilla Firefox Browser in Private Mode by default

Open Firefox
Type about:config in address bar
Press Enter
Type browser.privatebrowsing.autostart in Search:

Right click browser.privatebrowsing.autostart and click Toggle (value will change to True)

Restart Firefox to apply settings

Alternate method

Click Menu


Click Options
 
Click Privacy

Click Drop Down Box next to Firefox will: 
 

Choose Use custom settings for history
 

Select checkbox Always use private browsing mode
 

Click OK to restart Firefox