So on his trait "A bloody path to find my son" when the first character is KO'd I choose another. I had always thought I would keep choosing after the second character was KO'd but now I'm not too certain. How many characters can I choose in a game for this trait?

I'm certain that the intent is to keep choosing, but I see your point.

I thought it was a recurring "bloody path", but I see that there's an interpretation that the "path" has only two steps on it - a first and "another".

WK?

Reads like the second and third statements are cyclical in reference to "that character". Might've been more clear to reference "the chosen character".

This. As written now, it basically seems to go like this:

"At the beginning of your first turn, choose an opposing character. Thanos' attacks deal 1 additional damage to that character. When that character is KO'd choose another opposing character. Thanos' attacks deal 1 additional damage to that character. When that character is KO'd choose another opposing character. Thanos' attacks deal 1 additional damage to that character. When that character is KO'd choose another opposing character." etc. etc.

If you (OP) were to take it as the literal reading that you are understandably seeing, technically you would choose the second character and get no benefit. Because the trait doesn't actually instruct you as to what that second choice does.

Unless you read it as it being another instance of the first choice, triggered by a different event (the KO'ing of the chosen character versus the beginning of your first turn).

TL;DR: The wording is sloppy, but your gut should be right.