I recently adopted a friends cat (adult) after he passed away.  The
cat was indifferent to me but has warmed up and now demands attention
from me when I return home from work, etc.

My question is this:  If I do anything other than pet the cat on her
head/back she will bite me and run away.  If I touch her paw or rub
her too far down her side she bites me.  Is there anything I can do
about this?

THanks
T

Stop petting her?

My mother's cat is very particular about how she'll permit herself to
be petted. If you don't do it right, this kitty will bite, and
sometimes she bites really hard! Some cats are mighty choosy about how
they want to be stroked. Others love any kind of attention and
affection. It sounds as if you've got one of the tetchy kind.

I adopted a cat who stayed with me for over 20 years. During
most of that time, she didn't tolerate being picked up. That
only changed in the last 5 years of her life.

If she's letting you pet her head, that's good. One of the
ways you can speed up bonding and create more trust is to 
pet her on the bridge of the nose, above the tip, and up
between the eyes. Also on the "cheek". The latter area has
scent glands and she will appreciate your willingness to 
have her scent on you. The former area, at the bridge of
the nose, is an area that cats share with those they
consider to be "family".

My cat used to get ecstatic when she would sit facing me
and I would rub only that area with the back of my finger
and my knuckle. She especially loved the knuckle, and would
go into a frenzy of rubbing the bridge of her nose into my
knuckle when, instead on my rubbing her, I held my knuckle
still and rigid, pressed into that indentation in the bridge
of her nose. She would purr VERY loudly while this went on,
and seemed to have no end of enjoyment from it.

Doing that for extended periods will have her feeling like
she's been rolling in catnip, and should lead to bonding
that will make her more permissive in other areas.

sublime1-ga

I liked reading Sublime1's comment. It made _me_ feel like purring!

Tomster1957...

Your cat's reaction is not that unusual. My cat ( that I've had since
he was 5 weeks old and is now 4 years old )still bites me if I pet him
or brush him too much. At times, he's bit so hard, he's drawn clood :(

Every cat has a different personality, so as you get to know your new
kitty better, you'll become more accustomed to her tolerance levels
and personality.

Enjoy your new addition :)

Steph53

My cat does that too but I just keep petting her, unless she runs
away.  I think she actually does like it and we just play around...
she kind of bites me and grabs me with her paws and I keep petting her
and it's a little game.  So it may not be a mean bite but a
start-to-play bite.  You can get some scratch and bite marks but not
serious ones.