Discovered: A King in Rags
Per yesterday's patch notes, this quality should unlock "if you’ve played at least one full seven-round run with the King of Motley on your team". 7 rounds would make more sense than the 5 listed here (and would match the quality-change text). The patch notes don't mention the number of accomplices, though, so it may be possible to unlock this with a smaller team as long as the King is one of them.
The quality might also be unlocked by <a href="/wiki/Separate_from_your_accomplices" title="Separate from your accomplices">Separate from your accomplices</a> and/or <a href="/wiki/Take_a_break_from_the_game" title="Take a break from the game">Take a break from the game</a> under similar conditions.
However, I think I can confirm that it isn't unlocked by <a href="/wiki/Claim_a_trophy!" title="Claim a trophy!">Claim a trophy!</a>: I won my first 7 games with the King, and got the Distinction of Motley but no D:AKiR. (That would make sense since <a href="/wiki/Sit_and_wait_for_the_King_of_Motley" title="Sit and wait for the King of Motley">Sit and wait for the King of Motley</a> is locked if you've already got the Distinction, so there's no need to give you D:AKiR to unlock that branch.)
Edit: Oh, but I suppose it could be unlocked by Claim a trophy! if you didn't have the 9 accomplices necessary for the Distinction ...