This is a tough one. I kind of feel like its the father's responsibility to tell his family. Given the current issues you are having with the bio dad, I'm not sure I would want to be contacting his family around him. That said, I'll tell you 3 stories of different friends:
My 17 year old cousin had a baby and never asked bio dad for child support and never had anything to do with bio-dad or his family. Actually kind of blissful for her anyway to not have to deal with an "ex" all through her daughter's childhood. No shared holidays, no visitation. When her daughter grew up she found her bio dad and they have somewhat of a relationship but not much. This ship may have already sailed for you since you're seeking support anyway which will entitle him to visitation and I'm not sure it's the best route for the child.
Story #2, ok not my friend, someone that posted here on mamapedia. No interest from bio dad and family then when baby is not quite 2 yrs old, they are wanting to take him for overnights, court is ordering visitation and the little one is scared - they are strangers to him. That would break my heart.
Story #3, a friend of mine is the paternal grandma in this situation. Her son denies paternity during the pregnancy and for some time after (even after DNA tests done - LOL - talk about denial). My friend decides to believe the young girl and her parents and shows up at the hospital with gifts and flowers to see her grandchild. She has visitation once a week with her grandchild - loves him to pieces. Ultimately, talks her son into counseling and helps him accept the little boy. Happy ending.
So, all in all, I think it's better if you try to have a relationship with paternal gramma. You will know pretty quickly if she is going to be thankful that you contacted her or if she's going to be adversarial. I think it's worth a shot. Better that she is in his life from the beginning than coming back confusing him when he is a toddler.Good luck to you and congratulations on your soon to arrive new family member!