One Piece's Marines host some of the world's most dangerous combatants, leading the charge on behalf of the World Government against their main foes: pirates and the Revolutionary Army. Some Marines take a more hardline stance against these enemies than others. Vice Admiral Monkey D. Garp, Vice Admiral Smoker and his partner Captain Tashigi are a few major Marine characters who have collaborated with pirates in times when it was necessary. Others, like Koby, continue to be on good terms with Luffy personally. At the end of the day, however, the battle lines are usually firmly drawn. All the aforementioned characters remain loyal to the Marines and have resolved to look past personal ties if need be.
Kuzan, also known as Aokiji, is a bit different. He is one of only three canon characters to resign or defect from the organization and align with forces the Marines consider hostile (the others being the late Jaguar D. Saul and the Supernova X Drake), and as a former Admiral is the highest-ranking Marine ever to do so. After fighting with now-Fleet Admiral Sakazuki over the position on Punk Hazard, Kuzan resigned from the force after his loss and is now allegedly aligned with the Yonko Blackbeard. The latter is considered at this point in the story to be a paramount threat to the World Government, and therefore Kuzan's about-face begs many questions. Here's what we can deduce so far.
Aokiji vs Sakazuki
Most of Kuza's airtime in One Piece has been in his role as a Marine Admiral, and of the three members holding that rank at the time, he was by far the most benevolent. Although still capable of following orders to the point of begrudgingly freezing his friend Jaguar D. Saul to death, and fighting fiercely for the Marines at Marineford, Kuzan found many of the orders he was given as a Marine in conflict with his ideals. His biggest enemy in this sense is Sakazuki, who will kill even Marines that abandon their duties and has no qualms about murdering innocent civilians to achieve the World Government's objectives, as in the destruction of Ohara. He defied orders by allowing the sole survivor, Nico Robin, to escape with her life. He also presumably protected the Straw Hats from being fully blamed for the events at Enies Lobby.
In general, Kuzan was a rare Marine, along with his good friend Smoker, with a deep sense of morality instead of a blind devotion to the World Government. Since losing to Sakazuki despite being nominated for Fleet Admiral by the outgoing Sengoku, Kuzan likely felt that Sakazuki would push the force in a dangerous and objectionable direction. By aligning with Blackbeard, as well as allegedly other elements of the underworld, he has decided that his wishes are counter to the World Government and is now among the greatest individual threats working against them. This is due to the sheer power of his Ice-Ice Fruit along with the knowledge of the World Government's inner workings that he's attained up to his resignation. But the question remains: what exactly are his intentions?
Kuzan's Goals
The one-time Kuzan appeared in-person after the time skip, it was to save Smoker's life when the latter was nearly killed by Donquixote Doflamingo. Smoker, who had heard rumors of Kuzan's less-than-savory affiliations, was concerned about his friend, but Kuzan reassured him that he was still the same. If that's indeed true, then Kuzan's sense of justice remains intact, and he has simply decided that the World Government can't facilitate the moral changes he believes are necessary.
It's no secret that Kuzan has aligned himself against the World Government and Sakazuki, but it's unclear exactly what he wants to change. If he was seeking nothing less than the absolute dissolution and destruction of the current world order, he might have aligned himself with the Revolutionary Army. Instead, he has been collaborating on some level with Blackbeard, who the Revolutionary Army despises almost as much as the World Government. This seems odd. Blackbeard's destructive and chaotic personality doesn't meld smoothly with Kuzan's known beliefs.
Indeed, it's highly likely that Kuzan intends to use Blackbeard as an ally to further his own objectives, and only conditionally loyal to him. Jesus Burgess, one of Blackbeard's closest confidantes, suggests exactly that, telling his captain that Kuzan can't be trusted. Blackbeard countered by saying the same reasonably applies to another crewmember, Shiryu, once the Head Jailer at the World Government Prison Impel Down.
A potential truth is that Kuzan, who was unable to defeat Sakazuki, decided he couldn't challenge the Marines or World Government on his own. The protection of a Yonko like Blackbeard is extremely useful in this regard. The Yonko are so formidable that the Marines and World Government are extremely cautious when it comes to attacking their territory or allies. Blackbeard is probably the most dangerous Yonko due to his possession of two Devil Fruits, and the most likely to break this détente. This contrasts with the relative isolationism of Big Mom and Kaido, who each run their own countries, as well as Shanks's pragmatism.
Taking all of this into account, it's conceivable that Kuzan is using Blackbeard, but will not stick by his side long-term. How he behaves during the Straw Hats' inevitable conflict with Blackbeard will be an important indication of his true intentions. For now, it's clear he's playing the waiting game.