The Condensce is the black queen of the session though, right? Doesn't that mean the players have to defeat her to get the planet for her to rule over? Or do they just need to kill the king?

The Black and White kings have been killed already. The Black Queen’s defeat isn’t required to win a session. Right now the Alphas need to place the beta kid’s battle field at the center of Skaia, request the Genesis Frog from Echidna, and place that frog at the center of the battlefield and the new universe will grow.