- He is a distant family.
- He is one of my distant family.
- He is one of my distant family members.
The first is simply wrong; he is not a family (which would be a group of people), he is a family member or better, a relative. The second and third examples are not incorrect, but feel awkward to me.
- He is a distant relative.
- He is distantly related to me.
- He's some sort of cousin of mine.
- He's connected to my family.
- He's a distant connection of mine.
The first of this group is the simplest and probably the best. It is fine in either formal or informal writing. If "He is" is changed to "he's" it becomes a bit more informal, but not much. The second is a minor variant. The other three are a bit more informal, and deliberately vague about the nature of the relationship. By being vague, they suggest it isn't very important to the speaker, and thus that it is distant emotionally as well as in the family tree. All are natural English.