James Tiberius Kirk, whether in the Prime Timeline, the Kelvin Timeline, or even the Mirror Universe, is a maverick who insists on everything happening his way. Unfortunately, this often means Enterprise crew members are killed, Starfleet is disgraced, and other people have to clean up his mess.
The Kobayashi Maru test at Starfleet Academy is designed to see how a particular candidate would fare given an unwinnable situation. In the Kelvin timeline with Chris Pine as Captain Kirk, Kirk used any means necessary to win and make himself look good in the process. Several problems arose from his cheating behavior, namely that his maneuvers aren't applicable in future situations, not to mention wrecking the grading curve of other potential officers.
Kirk's behavior then flew in the face of protocols and regulations to the point of jeopardizing his crew, simply because he wanted to win with impossible foreknowledge. He showed his character that day by revealing the sort of captain he would later become - reckless and dogmatic.
His penchant for never forcing his crew to take a risk he wouldn't take himself is why many believe Kirk is the best captain, as Kirk was often the first person to beam down to a planet on an away mission. However, this is something other captains rarely do for the simple reason that if something happens to the captain while they're also with the first officer, the crew will be thrown into unmitigated chaos until replacements are found.
It takes a great amount of vulnerability in a leader to not desire to collect the glory for themselves, but rather rely on the judgment of others, especially when bulldozing through interactions with an alien species isn't an option.
On missions involving first contact with an alien species, more often than not Kirk broke the Prime Directive. Granted, he's sometimes under orders to influence the culture in some way, deliver an ambassador, but more often than not such as on a mission to Eminiar VII, Spock briefs the captain on establishing a diplomatic relationship and he stops a war in progress by force where he had no right to do so.
Kirk was supposed to honor Starfleet's instructions, but because the ambassador ordered him to enter orbit, most of the crew of the Enterprise was almost killed. Meanwhile, the civilizations on Eminiar VII were profoundly altered. Even if Kirk wanted to do the right thing at a time of strife, this situation was why the Prime Directive was established in the first place.
It's a commonly shared inside joke in the Trek community that if a crew member is wearing a red shirt, they're fodder for whatever alien of the week that Captain Kirk and his crew will face. That's because Kirk lost a lot of crew members, though they were often nameless and barely given any consideration.
This isn't to say Kirk was never held accountable, both by his own conscience and the Admiralty when he lost members of the Enterprise going into dangerous sections of uncharted space, but the fact remains many missions involving hostile aliens (like Rojan destroying members of the landing party in cube form) became notable for how many lives were lost.
Whether he's dismissing McCoy's former flame, delighting in not knowing what his own romantic partner Lieutenant McGivers actually does on the ship, or complaining about Spock's parents, Captain Kirk has a way of being insensitive around even those closest to him.
Luckily for Lt. McGivers, she finds a more sensitive lover in Khan. That's right - Kirk's eventual arch-nemesis is more appealing than he is because at least he pretends to care about her feelings and what her duties are.
Whether it's due to the acting style of Shatner or just because Kirk is caught up in the moment, he often lets himself be carried off by his emotions and then makes important decisions around them. More often than not, it involves the Klingons, with whom Kirk has a particularly tempestuous relationship.
In one incident, his eye for women and his personal problems around Klingons gets the better of him, and soon he's having Scotty manufacture firearms for the people of a peaceful planet, ultimately destroying their lifestyle and getting several people killed.
Whether it's violating General Order 12 or General Order 15 dealing with Khan, or inventing General Order 24 to suit his own needs, to paraphrase Spock, "he's well aware of the protocol, he simply doesn't give a damn." That brazen approach to Starfleet regulations is what made Kirk's captaincy so impulsive and selfish.
Disobeying orders in the name of getting the job done, no matter how, doesn't account for the cost, and too often the cost is too high. Going with his own gut on things is why promising young captains like Will Decker have to die.
Other captains like Pike, Janeway, or Picard choose to rely on the knowledge each of their crew members brings to a given situation. Kirk, by contrast, tended to rely on his own abilities, whether they be tactical, physical, or even romantic.
Where other captains appreciate the skills and perspective of their crew and trust their judgment, Kirk ultimately decided everything from battle maneuvers to diplomatic relations by trusting his gut instead of getting a consensus from trusted fellows.
Whether it was Yeoman Rand, Deela, Drusilla, Miramanee, or Elaan of Troyius, one of Captain Kirk's unmistakable traits was being a "ladies man" known to romance a different "woman of the week". Kirk's effect on women of any species seemed undeniable, and he often succumbed to lustful urges even if they were with sworn enemies.
Kirk's habitual romances have earned him a reputation as a womanizer, and compared to the amount of intimacy afforded Captain Picard, Janeway, or Sisko, it's earned whether he (or the writers) intended it or not.
While it seemed at first Captain Picard would have big shoes to fill taking over Starfleet's flagship, his mature and dignified air was a breath of fresh air given how much Captain Kirk used his fists to save the day.
Kirk often cracked skulls rather than employing diplomacy in situations that would have called for it. His manner of first contact with aliens almost always ended in a way that could have been salvaged had he considered how to approach them on their own terms. In several encounters with Klingons, he decides to fund arm races that favor the Federation.