How open should I be with my PhD supervisor about the problems in my personal life?
I think your vagueness is entirely appropriate and urge you to continue it. As long as your supervisor knows that there are difficulties that cause you to miss deadlines occasionally and as long as you feel she is supportive, there is no reason to say more. It would be a burden for her to know some of these things and yet, she isn't in a position to help you, other than to be understanding and supportive of your studies. 

But you should seek some sort of professional help elsewhere, perhaps at the university if it has a counseling office open to you for this, perhaps elsewhere. 

Also, sharing too much of the problem can start to sound whiney and like you are making excuses for things. 

Keep it light. Keep your relationship to your supervisor professional. Seek personal counseling elsewhere. And good luck.