Relationships
Being the villain of the game, Kuja interacts quite often with people -- but usually in a negative way! Spoilers ahoy.
Zidane
When these two first meet, it seems like this is the standard case of hero meets villain, thus they must hate eachother. However, it is clear that Kuja is interested in Zidane in particular, and the reason isn't revealed until deep into the game. Zidane's alignment with Princess Garnet interests Kuja in wrecking more havoc upon Gaea than he had even planned. For more information about the complex relationship between these two, try the Zidane section -- it is chock full of spoilers!
Brahne
The beautiful (hah) elephant lady. Kuja hates her with a passion, mainly because she's hideous, but that's all the more reason for him to manipulate her. He offers her weapons like black mages to use to take over the world. Part of his mission is to get Gaia to break out into war, and well, he uses Brahe as his means for doing so. Brahne is too blinded by greed to see that he is just using her as a pawn.
Garnet
Kuja extracts eidolons from Garnet, he corrputs and kills her mother, and he destroys her kingdom. Is it any small wonder that Garnet hates him? Despite her disgust for him, Kuja shows special interest in her that almost borders on romantic. Afterall, she is one of the few attractive characters in the game -- and we all know how much Kuja appreciates beauty! For more information speculation, try the love section.
Vivi
Out of everyone in the game, Kuja is cruelest to Vivi. Vivi is a black mage, and Kuja is the one who knows how to make black mages. Kuja is known to taunt Vivi with his recipe ingredients and hurts Vivi by controlling his black mage friends. Nevertheless, Vivi is determined to stand up to Kuja so that he can write his wrongs.
Zorn and Thorn
Brahne's little jesters join up with Kuja once she is... out of the picture. They're pretty annoying and talk in odd patterns, but they're quite useful little henchmen to Kuja.
Garland
Talk about a complicated father/son relationship. Although Garland isn't Kuja's biological father (Kuja is created, afterall), he is the one who created Kuja in order to save Terra. Instead, Kuja is hell-bent on rebelling against Garland, and ultimately he causes his demise. The only problem is that as Garland dies, he reveals to Kuja that he was created with a limited life span, since Garland worried he could potentially become too powerful. Ooops.
Mikoto
Mikoto is basically a sister to Kuja, and even though he does some horrid things, she looks up to him. He was the only genome to seek independence and freedom, and that's inspiration to her and the other genomes. Jeez, who'd have thought Kuja could be a positive inspiration?