Kadaj

"So what if I'm a puppet? Once upon a time, you were too!"
—Kadaj to Cloud

Although Tetsuya Nomura wanted Sephiroth to appear in the film from early in production, a secondary villain just as evil and cruel was needed, if not a villain more evil due to the film's limited timeframe. It is for this role that Kadaj was made.
Kadaj

Kadaj embodies Sephiroth's cruelty and insanity, and as such is very similar to him in some aspects. Nomura has stated that Kadaj wasn't given an exact age, but was given a young appearance and made the leader of the Remnants to give the sense of a psychotic, unstoppable child. Kadaj's voice actor Shotaro Morikubo in the original Japanese version of the film, helped shape his character as a lost and dazed youth with an insanity streak - Nomura describes him as "talking quickly, then speaking slowly or breaking into laughter for no apparent reason". Morikubo joked that Kadaj's lines were thus difficult to get right, and it was only after failing to get them right and "losing his cool" that the desired effects would be achieved.

Several of Kadaj's scenes, including his death with the involvement of Aerith and his conversations with Rufus, were last-minute additions originally cut and added back in to expand on the plot and his character, and the death scene was meant to give a sense that he had found some sense of relief. As well, the staff invisioned him as a reminder of what Cloud once was - a puppet to Sephiroth's will - which led to the inclusion of the line in which Kadaj reminds Cloud of this. This line in particular endeared Kadaj to the staff, which led to the development of more scenes featuring him.
edit Story

Kadaj initially appears to summon Shadow Creepers to attack Cloud Strife while he is out riding his motorcycle. After speaking to one of the Turks over the phone, he recalls his monsters and sets off to find Rufus Shinra. Arriving shortly after Cloud leaves Rufus, Kadaj reveals that the plague Geostigma is caused by the remnants of Jenova in the Lifestream, and that those infected are part of a plan for a second Jenova Reunion that will recreate Sephiroth. However, Geostigma alone isn't enough, Jenova's remains are needed, and Kadaj believes Rufus knows where they are.

With a group of infected children brought to the Forgotten Capital, Kadaj proceeds to tell them that Geostigma is caused by their bodies rejecting Jenova's influence (which ironically, is correct), and that her presence makes them brethren. Kadaj has the children drink from a pool of dark water, activating the latent "reunion instinict" in their Jenova cells. Cloud arrives afterwards and battles Kadaj and his brothers. Outnumbered and weakened by his own Geostigma infection, Cloud is defeated but is saved by Vincent Valentine. It is at this time that Vincent reveals to Cloud what Geostigma is, and the impending rebirth of Sephiroth it could mean, although he doesn't yet reveal Kadaj's connection to Sephiroth.
Kadaj's CG render for the 10th Anniversary of Final Fantasy VII'
Kadaj weilding Souba.

With the reunion instincts of the children to guide them, the Remnants go to Edge, where Loz and Yazoo battle the Turks and Kadaj summons Bahamut SIN to attack the city. During the attack, Kadaj confronts Rufus again, and confirms that Sephiroth is about to be reborn. After Cloud and his friends defeat Bahamut SIN, Rufus reveals Jenova's head in a box concealed under his cloak. Casting the box off the building where they're standing, Kadaj leaps after it and catches it. The ensuing chase between the Remnants and Cloud along the highways into Midgar apparently kills Yazoo and Loz. Kadaj gains the lead and stops at the Sector 5 Church to inspect Jenova; he is promptly horrified at what is left of her and cries. Cloud bursts in, and an enraged Kadaj attacks him. However, Aerith calls a geyser of healing water to erupt from the flowerbed, wounding Kadaj and causing him to flee. The water also cures Cloud of his Geostigma.

Cloud confronts Kadaj outside the ruins of Shinra Headquarters. Kadaj is unaware of what to do with Jenova's remains now that he finally has them; when Cloud points this out, he attacks. Now that he is cured and back to his full strength, the battle ends with Cloud as the victor and Kadaj hanging from the edge of a tower. With no other option, he leaps into the air and absorbs Jenova's cell into his body, leading to the recreation of Sephiroth moments later. After Sephiroth is defeated, Kadaj is reformed from his dissolving remains, and he attempts a half-hearted attack on Cloud but collapses into his arms, mortally wounded. Weakly calling him "brother", Cloud holds him while Aerith speaks to Kadaj from the Lifestream. Believing her to be Jenova, Kadaj calls her "Mother" and happily dissolves into the Lifestream with her guidance.
Spoilers end here.
edit Strength
Kadaj performing a Limit Break?

Kadaj is a highly skilled swordsman and is able to battle Cloud competently one-on-one, although he is defeated. When Cloud is weakened due to Geostigma, Kadaj is able to match and defeat him, and he is evidently strong enough to defeat both Reno and Rude by himself, as they are incapacitated when he confronts Rufus without his brothers. Kadaj also has access to Materia in the film, and uses it both to summon Bahamut SIN and to cast Matra Magic. Even without the aid of Materia, Kadaj is able to summon Shadow Creepers at will. In Advent Children Complete, Kadaj is able to infiltrate Healin Lodge by transforming into a black mist.

Upon walking into the water at the Forgotten Capital, it turns dark and the Jenova cells within the children become active when they drink it. One might assume this is another power accessed by Materia, but this is incorrect; the Materia orb in Kadaj's gauntlet does not glow when he uses this power, as it does whenever he otherwise uses Matra Magic. It has been speculated that this may be some form of Limit Break due to the way Kadaj glows when activating this power.