Kingdom Hearts thirsts for hearts... a thirst only your Keyblade can slake.
Along with Xemnas, Saïx appears to be one of the most "Nobody-like" of the Organization. Despite claiming to remember what possessing a heart was like, he behaves mostly as a cold, emotionless, and efficient operative, even in the direst of situations. He is very aware of how emotions are unreal to them, unlike other members who imitate their emotions.
However, in Kingdom Hearts 358/2 Days, it seems he values his friendship with Axel from their time as humans, indicating that he believes that Nobodies cannot even try to genuinely show emotion or form a real friendship without having had one with another beforehand, thus having memories to fall back on.
According to Axel, friendship between them weakened due to how Saïx changed too much after becoming a Nobody, leading to Axel's ignoring him for Roxas and Xion.
The only thing Saïx seems to "care" about is acquiring a heart and, by extension, Kingdom Hearts. More than any other member of the Organization, Saïx is portrayed the most dedicated to their goal, repeatedly stating his desire to "truly, finally exist". Even his last words were to Kingdom Hearts, sadly asking for his heart. Though this doesn't make him blind to Xemnas' plans to use Kingdom Hearts for more power, as he had originally plotted with Axel to overthrow Xemnas to regain their hearts.
Saïx can also be quite cruel, claiming to know very well how to injure a heart. He had a keen insight into manipulating others, such as when he toyed with Sora's emotions by having him beg to see Kairi, only to refuse his request. He also sent the "useless" members to Castle Oblivion under the guise of Xemnas' orders to clear the way for his and Axel's plans, moving up the ranks while eliminating any competition or rebellion.
However, his usual calm exterior belies a more savage nature. In his final confrontation with Sora under the heart-shaped moon of Kingdom Hearts, he becomes a berserk force of destruction. Unlike his calm and reserved manner, in battle, he roars, shouts, and attacks relentlessly with no sense of self-control and portrays a psychotic behavior under the influence of Kingdom Hearts. "If I had a heart, this would be where I die of laughter."