I think you've partially answered your own question... Lister cannot possibly have grown up with his biological family for multiple reasons:
1) He was abandoned under a pool table in Liverpool- any biological relatives he did have wouldn't even have known of his existence, and to that end:
2) He has fewer potential biological relatives than your average person, as He is his own father.
This means that the "father" he mentions in S02E03, "Better than Life" must be an adoptive father. We know that his adoptive father raised him until he was six, at which point he died (unknown causes).
Lister: I remember when my dad died, you know. I was only six. I got loads of presents off everyone like it was Christmas. I remember wishing a couple more people would die so I could complete my Lego set. My grandma tried to explain, you know.
This also establishes that his adoptive father was single, as Lister never seems to mention a mother being present, and her absence would explain why he would go on to be raised by his grandma.
As to the question of: "Was Lister related to the "grandma" who raised him", the answer there is also a no. We know that, as she was the mother of Lister's adoptive father, she couldn't have been related to Dave.
[Edited to add]
Another, more compelling note is the fact that Dave can only truly have one other biological Grandmother. Being his own father, the only actual grandma he can have comes from his mother's line. Given that his mother is Kristine Kochanski, it stands to reason that, had he been raised by Kochanski's mother, he would have grown up thinking he was Kochanski's brother.