Personality

Cait Sith's personality is most accurately defined as paradoxal. Despite appearing cheerful, friendly, and a bit frivolous on the surface, his happy-go-lucky personality, as well as his cat-and-mog physique, is actually a clever stratagem to disguise his very human character puppet-master.

When the party first meets Cait Sith, he's shown to be quite merely a goofy fortune teller, one who just happens to be more than willing to lend his services. However, his determination and pushiness to join Cloud and the others gives them cause for suspicion as to his motives for wanting to tag along. Despite this, he's quickly accepted into the group and lends his friendly, bouncy disposition to the journey.

Underneath it all, Cait Sith harbours a secret identity as well as a secret mission, both of which he must take pains to keep hidden. Even after being accepted by AVALANCHE, he knows he must put himself into or create situations that will perpetuate his appearance of genuinity. To do this, he brings to light the suspicions of others in order to throw uncertainty off of himself.

CaseOne

(The party stumbles upon the Turks in Gongaga and realise they've been followed.)

Cloud: "They followed us...... But there weren't any signs of it. Then, that means......"
Other Party Member: "A spy!!"
Cait Sith: "Oh no! Now you're gonna blame me 'cause I'm new." ∗

Obviously, Cait Sith is to blame, but being the first to raise the suspicion makes him seem more likely to be innocent rather than guilty. He's just attempted to psyche the other characters into into disregarding Cait Sith himself as a possible suspect through showing that he himself harbours (even if he doesn't really harbour) the same fear of a spy being in their midst.
∗ Line not game-accurate, as I forgot to jot it down whilst playing. But you get the picture.

CaseTwo

(The party enters Nibelheim to find it not having been burned down like Cloud previously said it was.)

Cait Sith: "Cloud, did you just make up that story to gain our support?"

An ironic comment, considering Cait Sith himself has done the same thing as he's accusing Cloud. Cait Sith has "made up" a lot of things himself to gain the party's support. Aside from his identity as a fortune-telling machine, he fabricated a certain prediction to get on everyone's good side—but in particular Cloud's—and he later created another reason to gain entrance and stay in the group.

Soon enough, Cait Sith's role as a Shinra spy is exposed hand-in-hand with his hidden nature. Caught red-handed, he has no choice but to admit he's a double agent working for Shinra. However, he's resolved to continue the journey, using whatever reason he can give. Whether there's truth in his reason or not, Cloud doesn't believe him, and informs Cait Sith of his personal philosophy on those like him: "once a spy, always a spy".

It's at this point that the ugliness in Cait Sith's—and by extension, Reeve's—nature is revealed. Cait Sith has a backup plan, one that though he regrets to execute he feels he must nonetheless. Willing to do whatever it takes to follow his orders, Cait Sith has Barret's daughter Marlene kidnapped and held as a hostage. He threatens to harm her if Cloud and co. don't do as he says. As things work out, we don't know whether or not Cait Sith would have actually harmed Marlene, but an intent to harm a child did exist. And that, in Cloud's own words, is low.

I didn't want to do this... using dirty tricks and taking hostages... But this is how it is... no compromises.

Cait Sith

After this debacle, all trust in Cait Sith is lost, and he knows it. If Cait Sith cares to get back on AVALANCHE's good side, he knows it'll have to be up to him to do it. Thankfully for Cait Sith, a chance happens upon him quite soon afterward. If Cait Sith and Cloud's group have one common goal, it's this: to get the Black Materia before Sephiroth does. Travelling to the Temple of the Ancients, the party realises that acquiring the Black Materia currently held within the temple would mean certain death to the person retrieving it. Knowing there's no other way for the party to proceed, Cait Sith valiantly offers his artificial body to do the dirty job. Despite being remotely controlled by Reeve at Shinra Headquarters, Cait Sith is somewhat of his own being: such a sacrifice is a huge commitment for him to make, especially when it concerns helping out his supposed enemy.

Don't forget about me even if another Cait Sith comes along.

Cait Sith

As strange as it may seem, Cait Sith is quite emotional about his decision, and even the party feels uncomfortable about it. Cait Sith sincerely thanks the group for believing in him, even though he was a spy. In his final monologue, Cait Sith fumbles over his words with obvious nervousness. He even admits, much to his personal embarrassment, that he's essentially become a saviour of the planet—a label he probably thought he'd never have applied to him. In his first show of inner strength, Cait Sith completes his self-bestowed sentence, turning the Temple into the Black Materia and crushing his body in the process.

Cait Sith no.2 shows up and seems the first reborn, having retained all of Cait Sith no.1's physical stats and memories. Something, however, has begun to change in Reeve, and thus in Cait Sith. Being a part of a good cause seems to be eating away at the duo, not to mention the fact that living with a group of people for an extended period of time, even if they're supposed to be the enemies, starts to breed a certain level of affection and sympathy. When Tifa is captured by Shinra, Cait Sith surprisingly shows up to help, saying in his defense:

Let's just say I'm against capital punishment. Besides... I hate this broad [referring to Scarlet]. Come on, we gotta help Tifa. I'll keep watch at the entrance.

Cait Sith

Whether or not those excuses are true, the underlying message is that Cait Sith doesn't want to see his comrade get hurt. And then, after hijacking the Highwind and saving Tifa, Cait Sith proceeds to make an even more significant statement:

Barret: Dammit! Why didn't ya tell me you could fly an airship!?
    I thought... Tifa was gone for good.
Cait Sith: Sorry, but I had to do something to trick the enemy. Come on! Everyone is waiting!

Perhaps without realising it, Cait Sith has just confirmed his true allegiance to Cloud and the others by referring to Shinra as the enemy. Cait Sith doesn't stop there, giving the party access to secret meetings via teleconference-style eavesdropping, as well as revealing the locations of the Huge Materia to keep them at the same level as—or ahead of—Shinra.

Unbeknownst to the others, this is a dangerous thing for Reeve to have Cait Sith do, as it would and eventually does come back to him as the informant. Even when he's discovered, Reeve gives out as much information as he can before he's arrested by Scarlet and Heidegger and expelled from Shinra. This doesn't stop him, however, and he continues his journey as Cait Sith with Cloud and the others.