This is the way that I beat it:
Research, production and military bonuses are important in this mission. Don't feel that you have to stay with the same political party and ability bonuses for every mission in the campaign. Try to pick the best bonuses for that mission.
1) It's very important that you don't cripple your Altarian allies by taking too many of the planets. Take the sectors on the right side and let the Altarians get the ones on the right side. However, their cultural influence can cause trouble for you later so make sure that you don't let them get any of the influence resources.
2) Once you've established a line of colonies up to the battlefront, start building constructors in your lower colonies and attack/defense ships in your upper colonies. Set a rallypoint on top of the star closest to the Korx. Make this the destination of all your colonies once you have a ship in orbit. This will keep a steady stream of warships going to this planet and keep the korx busy. Make a planet in the lower right hand corner that's in a system with two colonized planets build freighters until you've built enough to max out your traderoutes. Send those freighters to the Altarians (no use trying to trade with the Torians; the korx will just blow up your ships and it's not worth the escort).
3) While the korx are busy trying to get rid of all your ships in the front line stars, build transports from further planets and start sending them up to the front line. As you get a sufficient invasion force, start taking out the nearby Korx planets. You'll probably have to destroy the conqured colonies to keep them from falling back into Korx hands. Use the constructors to build mining starbases and give you production bonuses. If they're along the trade route path, build the trading modules too.
4) Make sure to spend money on destabilization and espionage on the Korx.
I was going for terror stars, but I think that I ended up beating the mission before I built any.