No, "upbringing" would not be right. Oppvekstvilkår refers to the living conditions of the child; the environment that the child grows up in. For example in a rich or a poor family, in an well-educated or less educated family, in the inner city or on a farm, and so on.
My dictionary translates "oppvekstvilkår" as "growing up conditions", so I suppose that would be the best choice. But as native English speakers, you know better than me how that expression works in English.