Story
The first time your party encounters Kuja is at the end of disc 1. There he is known as Mysterious Man, and he is talking to Brahne and Beatrix about finding the citizens of Burmecia and attacking Cleyra. When Zidane and co. are discovered, Beatrix fights them and ultimately wipes them out. Afterwards, Kuja looks at them sprawled on the ground and remarks that Zidane may be a problem. He walks over to his silver dragon and makes a dramatic getaway.
Our next glimpse of Kuja is in Treno, where he is talking to the auctioneer. The auctioneer asks Kuja how Burmecia was, and Kuja says it would have been better without Brahne. The two seem to be arranging some sort of deal... I'm assuming that they're talking about black mages. At any rate, the auctioneer agrees to have them ready for when Kuja goes to Cleyra. Kuja asks if the auctioneer saw a pretty lady in the crowd, and then Kuja explains that he has almost captured her. This lady is Garnet.
Later, at Alexandria Castle, Kuja puts a spell on Garnet to make her fall asleep. He does this in order to extract eidolons from her -- powerful summons. He convinces Brahne that she can have a massive amount of power with them.
Although Kuja isn't seen for awhile after that, your party sure talks about him quite a bit. At any rate, his next scene is at the Iifa tree, where he taunts your party and watches as Brahne sets the powerful eidolon, Bahamut, on him. He gets away with only a scratch and laughs. Next thing you know, Kuja turns Bahamut against Brahne, and, well, she's dead meat.
Kuja next shows up in Alexandria, right before he sets Bahamut on the entire city. He makes a little soliloquy, talking about how overjoyed the people of Alexandria are because Garnet is their new queen, but how it will be all over once Bahamut ruins their lives. Needless to say, Bahamut does ruin their lives, and Kuja is the cause of it. When Alexander is called by Garnet and Eiko, it looks like Alexandria will be protected. However, a mysterious ship known as the Invincible appears over the sky and ruins Alexander. Unfortunately for Kuja, the Invincible doesn't capture Alexander like he thought it would, and he assumes that a man named Garland is the reason.
After running around a bit and trying to find Kuja, Zidane and co. find him at his home in Gaia, the Desert Palace. Here Kuja keeps everyone locked up in their own little rooms, and he calls Zidane and three party members of his choice to come complete a mission for him. He sends them to retrieve the Gulug Stone from a town called Oeilvert. He threatens to kill the remaining party members if they don't return with it. Once they return, Kuja takes the stone by force, and along with it he takes Eiko, successfully captured by Zorn and Thorn.
Kuja takes the Gulug Stone to a place called Mount Gulug, and Zidane and the rest of the party naturally follow. Once they catch up to them, they see that Kuja is having Zorn and Thorn extract an eidolon from Eiko. They end up failing because Mog, Eiko's moogle, protects her by going into a trance. This gives Kuja a new idea for power -- if he obtains a powerful soul, or a whole bunch of them, he can experience a powerful trance. He sets off, leaving Zidane and co. to Zorn and Thorn.
The next time you encounter Kuja is in his home planet, Terra. He captures the Invincible, and he tells Garland that he will take over. However, Garland tells him something he never knew -- Kuja's life span is limited, he is due to die any day. Needless to say, Kuja doesn't take this too well. Once he fights Zidane (finally!), he trances and tells the group that he is achieving this trance by using the souls of the people of Terra. He proceeds to wreck the entire planet.
At the end of the game, Kuja is still in trance mode, and you get to fight him again. He's significantly more powerful, and his goal is to kill everyone. Since he can't live, he doesn't want you to live either. But, you beat him, and so his goal is lost.
The last time you see Kuja is when Zidane runs into the Iifa tree after hearing his voice. He gets to talk to Kuja one last time, and Kuja admits that he was wrong. He says that he doesn't deserve to live, that he's useless to the world. He figures out the meaning of life, but he acknowledges that he's too late. Just as he and Zidane come to terms with each other, Kuja dies.