Kubik's story begins with the A.I.M. Cosmic Cube. A Cosmic Cube is a containment field for an enormous amount of energy from the universe of the Beyonders (plural, not to be confused with the Beyonder, singular, who is also related to Kubik). The Cubes can be used to fulfill almost any wish that their holder desires. A.I.M. created their Cosmic Cube for their own purposes, but it had many masters over the years, including the Red Skull and Thanos, most of whom tried to use it to conquer the world and/or universe. There are other Cubes, but this was one of the most popularly-used. A.I.M. also used a small sliver of their Cube to power the Super Adaptoid.
It turns out that each Cube is not just an object to be wielded by its master, but it also eventually evolves into a sentient being with the same enormous power as the Cube itself. The race of Beyonders, who created the possibility for the Cubes, are very interested in the idea of evolution and desire, and so the Cubes fulfill both of their obsessions: they bring their masters' wishes to reality, and they themselves evolve into beings that are fascinated by desire. The first known sentient Cosmic Cube was the Shaper of Worlds, who evolved from a Skrull Cosmic Cube. The A.I.M. Cube eventually evolved into Kubik. The other major sentient Cosmic Cube is Kosmos, who was created by merging the power of the Beyonder and the Molecule Man.
Kubik first developed sentience while being used by an A.I.M. scientist named Bernard Worrell. He used the cube to transform the entire Earth into a "technopolis." However, the Cube began to fight back against such ill use. The Shaper of Worlds appeared and tried to help the Cube, which had begun to grow tentacles and move around. Eventually, through the aid and good will of the Shaper, Captain America, and the Aquarian (who had earlier nurtured the Cube, and been aided by the Cube in turn), the Cube became calm and began to actualize itself as a truly living being. The Shaper agreed to take care of the newly-born Kubik, so as to avoid the growing pains and concomitant destructive period that the Shaper had gone through when he had first become sentient. The Shaper and Kubik went off to explore the universe.
Cloud and the Star-Thief
Eventually, Kubik was left to his own development. At one point he felt fear radiating from Cloud, a sentient nebula in space. She was worried because all the stars around her were disappearing, and she didn't know why. If they kept disappearing at the same rate, the whole universe would soon be dark. Kubik, having been influenced by Earth's heroes, wanted to help. He condensed all her gas molecules to human scale, then sent her to Earth to get help. Unfortunately there were complications and she forgot her mission. Eventually, with the help of the Defenders, she finally remembered, and they all traveled to space to meet Kubik, who at this time took on the form of a spiky planetoid.
Kubik was glad to see them: he missed seeing Earth folk. He said that he would normally just put the stars back in place, but for some reason he couldn't do it, so he needed their help. He then took on the form of Captain America, who had impressed him when he first started to become sentient, and joined them on their quest. He enjoyed playing the hero and using his powers to help in minor ways. It soon turned out that the stars were being taken by a comatose alien princess who called herself the Star-Thief. She had been wounded by a laser and now wanted to destroy all light. She was a powerful mutant and her intense will had somehow managed to take hold of Kubik's wish-granting power. Using that power, she was taking the stars and putting them in a pocket dimension. (This alien Star-Thief is not to be confused with either of the other two Star-Thiefs, Ditmal Pirval or Barry Bauman. Barry Bauman's story is actually almost exactly the same as the alien Star-Thief's, in that he was also a powerful comatose psychic who took stars and put them in a pocket dimension! Even the psychic projection of his face looked very similar to hers.) Eventually the Defenders managed to convince her to stop stealing the stars and Kubik returned them to their rightful places.
Challenge of the Super Adaptoid
The Super Adaptoid, who had a sliver of Kubik's energy, wanted to take advantage of Kubik's full power. He sent a call out, and Kubik came to earth. He now appeared in a humanoid, almost robot-like form, wearing silvery armor. The Super Adaptoid copied Kubik's enormous power and tried to banish him to a black hole in the Andromeda Galaxy. However, Kubik was unfazed by this development and removed himself from the black hole and returned to Earth. The Super Adaptoid was trying to take over the Earth. Kubik wanted to stop him and remove his power, but said that if he confronted the Adaptoid himself, it might destroy the universe. Therefore he contacted Captain America, who he knew was very resourceful, even if he did not have the power of other heroes. Captain America essentially managed to trick the Adaptoid into killing himself, and Kubik removed his power, then returned to space.
Creation of Kosmos
Kubik next wore yet another form, which he has retained since, as a pink humanoid with a large head and a glow in his chest and/or head. Kubik and the Shaper of Worlds confronted the Beyonder, explaining to him that the he and the Molecule Man were parts of an incomplete Cosmic Cube. While Reece seemed to grasp the truth in this, the Beyonder's massive ego did not let him face facts so easily. The Beyonder had previously thought that he was the most powerful being in the entire Marvel multiverse, but now he was being confronted with the fact only one of a group of powerful beings. They all did battle, and the Beyonder had to face up to the fact that he was not as powerful as he had thought. The Shaper explained more about the Beyonders and their role in making the cube, and their destiny as beings of evolution and desire. Ultimately, Kubik and the Shaper convinced the Beyonder and the Molecule Man to merge into a new Cosmic Cube. This Cube eventually "hatched" and become a new entity called Kosmos. (Reece's essence was sent back to Earth.)
Kubik then took Kosmos on a long journey and taught her to understand the nature of the universe. They transcended to a universal scale and viewed some of the various powerful beings of the multiverse, such as the Stranger, the Vishanti, various demons, the abstracts like the Inbetweener, and so on. He explains that while she thought she fought many of them in her incarnation as the Beyonder, the beings really only sent limited avatars of themselves to help her "transition to awareness," and she had never really fought them at their full power.
They investigated the riddle of the Celestials, pondering their origin and motives, as a way of better understanding the universe.
We later see them performing an experiment on the Fantastic Four, in which they alter their personalities and run them through a dangerous maze. The goal of this experiment was to see why Earth is such a nexus for the important events in the universe. Ultimately they find that the Fantastic Four are still very effective even when their normal advantages are taken away, showing the great potential of mankind. Kubik notes that in a sense humans have greater potential than the sentient Cubes, for humans can change and grow in response to adversity, while the Cubes are static in their omnipotence. (However, this seems to completely contradict not only the Cubes' basic raison d'etre to personally evolve as directed by the Beyonders, but also the exact reason Kubik and Kosmos are touring the universe: they want to learn about it to learn their roles in it.)
Confrontation with the Molecule Man
Next, instead of seeing Kubik experiment on humans, we see the Cubes as an experiment themselves, controlled by the even higher powers of the Beyonders. Several results of this experiment come together: Owen Reece, Kosmos, and Kubik. Kosmos and Kubik were still touring the universe, become more and more rarified. Reece was back on Earth, where he had been spurned by his lover, Marsha Rosenberg, and that loss let his dark side take over again. This dark side was still regressively furious at the Beyonder, and pulled the old Beyonder persona out of Kosmos from across the galaxy. Reece and the Beyonder battled, while Kubik (who had now fallen in love with Kosmos) desperately brought the near-comatose remaining essence of Kosmos to Earth. The Molecule Man, who had previously always had less power than the Beyonder, was now able to beat him. Kubik explained that while each of the Cubes theoretically had equal power, Reece's was actually greater due to the fact that he was human, with all the potential for growth that he previously found in the Fantastic Four, while the other Cubes were all abstract, "elemental omnipotents," and "nothing without their powers." The Molecule Man's good side, seeing Kubik's feelings towards Kosmos, takes over again, and he restores Kosmos to her true form. Kubik and Kosmos leave to once again explore the universe together.
Kubik has not been seen since.
Kubik's personality has evolved, as it is supposed to for a sentient Cube. At first he was fairly whimsical, and loved Earth and its superheroic denizens, especially Captain America. He became more analytical and "unfeeling" in his persona as he embarked on his program to learn about the universe in a more serious manner, and took on his form as a armored robot-like humanoid, but still retained his passion for Earthlings.
During the confrontation with the Beyonder, Kubik took a more aggressive role than the Shaper. The Shaper was mostly content to talk, while Kubik leapt into action and fought the Beyonder, almost crushing him. While somewhat out of character for his previous showings, this fits in their "family" dynamic. The Shaper is the oldest "brother," and more mature. Kubik is like the adolescent of the group, while the Beyonder is the youngest and most like a child, unable to control his unchecked emotions.
The later Kubik becomes more and more analytic, and seems to care less about Earth heroes, as seen in his experiment with the Fantastic Four. He has a passion for learning. But he seems to truly mature himself as he begins to tutor Kosmos. They even developed a relationship together, akin to human love. As Reece said, he "started human and became omnipotent [while Kubik and] Kosmos started omnipotent and are learning to be human." It is unknown why Kubik eventually apparently left Kosmos, since she later became the Maker and fell into madness and apparently death.
Kubik manipulates extra-dimensional energy to bend reality into anything he wants, within limits. The Cosmic Cubes are essentially wish-granting devices, and the sentient Cubes seem to have more or less the same kind of power. The Molecule Man, who shares similar power, has stated that the Cubes are not as powerful as Eternity or the Living Tribunal. Kubik himself stated that the sentient cubes' "power is as nothing compared to that of the Celestials...[whose power is] many orders of magnitude beyond our own. We could not possibly contend with such a being and hope to survive."
Kubik has been shown and described as having varying levels of power. When the Super Adaptoid copied his power, he was described as the "master of all reality" and that he could "basically run all reality." When Kubik considering fighting the Adaptoid, he said that he (Kubik) was the "only other being in the cosmos of equal power" and that "the resulting reality war would shred the fabric of all existence--extending into infinite dimensions and adjacent universes." He said that he "transact[ed] on power levels unimaginable to" the post-retcon Beyonder (because the Beyonder had not yet evolved into his final form). In describing the Cubes, he says they have a total mastery of their environment. Many of these quotes, however, may be understood within the context in which they are made: they have a total mastery of their environment in comparison to humans, for instance, but perhaps not in relation to beings such as the Living Tribunal.
For instance, in the other direction, Kubik has been described in narration as being a "minor omnipotent," far below the truly major powers in the cosmos. Kubik himself said that the sentient cubes are "insignificant" compared to the greatest powers of the cosmos. It is unclear exactly what powers he was referring to here, but just previously he was talking about abstracts like Eon, as well as other powerful beings like the Celestials and the Vishanti.
Here are some examples of the things he has done: Condense a nebula (Cloud) to human size. Steal thousands of stars and put them in a pocket universe (while under the influence of the Star-Thief), then bring them back again. (It was implied that he could potentially do this to all the stars in the universe.) Create a giant copy of Captain America's shield to protect the Defenders from re-entry into a planet's atmosphere. Easily teleport across interstellar space and travel between dimensions. Quickly escape a black hole and easily change its rotation and electrical charge. Alter his size to become many times the size of a planet (he did this both in the 616 universe and the Beyond-realm) and even hold a sun in his hand like a marble. Battle the Beyonder (post-retcon), absorb an attack in which the Beyonder threw hundreds of planets at Kubik, and eventually grow so that he held the Beyonder and the entire Beyond-realm in a sphere in his hand, which he said he could crush. Easily create a giant maze construct that was perhaps on the scale of a moon, filled with various technological traps.
When the Super Adaptoid copied his powers, he grew to giant size, created five billion human-size adaptoids, and created prisons that were inescapable even to the Black Knight's sword or Captain Marvel's energy form.