Should I finish the game in both chaos's before starting the doc, or can I just play it after finishing my high chaos playthrough. No spoilers please.

Your save file from Corvo's campaign isn't imported into DLC, so you can play them whenever. Just finish your current game and move on to Daud's story.

I know, but does the other ending affect my understanding of what's going on in the doc?

No. Corvo's decision regarding Daud will affect how you feel about the ending of the DLC, but it has nothing to do with understanding the story (or your chaos level in the main game, for that matter).