If something was a fluke, it means it happens by chance and without explanation. I first time I tried, I hit the target. It was just a fluke; I've never hit the target since.

The noun "fluke" is an event that happens one time only. It does not imply whether the event is good or bad, nor whether it is was caused by luck or skill. (Example: The army won the war, then they vowed never to fight again, so it was a fluke.) In your example, the event should be "being born". "having a family" is not a one-off event. So try forming your sentence again with that knowledge.

Both are understandable. Both should probably be in the past tense, "he moved up the ranks." #1 is not correct. In #2 I would use the word "through" instead of "by," "through the fluke of..." #2 is a little unnatural, though, for two reasons. The first because the usual context for the word is a phrase like "it was a fluke," or as an adjective in the phrase "fluke accident." Second, "fluke" usually means something weird, surprising, something so bizarre that you would never expect it to happen. "Having a rich family" is good luck, but it isn't really "a fluke." Here are some more typical examples of use. "That was an unbelievable goal. That had to be a fluke." "No, that was no fluke. She's really that good." "In a fluke accident, the child was locked inside a school bus overnight, but was rescued unharmed."

1. any family 2. his own family