I don't think Slaine cares so much about ruling the empire so much as reforming it. He is a really idealistic guy who didn't even grow up on Earth, so it is not surprising that he doesn't really feel anything for Earth's side of the war. On the other hand, he grew up on Mars pretty much as a slave, and he bonded with the culture and society there even though he was mistreated. He definitely has an affinity for Martians in general, and the suffering of the normal populace. I think a large part of this is also due to how he sees Martian society through the lens of the princess, and perhaps he sees himself as fulfilling her dream in her stead.
Either way, this is probably going to end badly because he doesn't have the experience or the real support to see the dream through, but simply actively moving forward with his plan will probably shake things up big time. In the end, he probably just wants to see a fairer and better world for the Martian people.