Bothering by the Book: Has the audacity to cite Starfleet regulations at Admiral Kirk. After the Enterprise is nearly destroyed because Kirk ignored her advice, Kirk orders her to keep on citing regulations at him.
She was intended for a major role in The Undiscovered Country, but between casting difficulties and objections by Gene Roddenberry over an established character becoming a traitor, her name was simply used as a placeholder for Kim Catrall's character, Valeris.
The Stoic: She's a Vulcan, so it comes with the territory.
Not So Stoic: She's young, so it comes as no surprise that she's often barely keeping a lid on her emotions. In a Deleted Scene, Spock tells Kirk she's half Romulan, but the canonicity of that has been debated ever since.
Dr. David Marcus (Merrit Butrick)
Armies Are Evil: Very ambivalent about working with Starfleet due to concerns with how they would use his research.
Properly Paranoid: Doesn't like working with Starfleet because of how the Genesis research could be abused as a weapon. Sure enough, the Genesis project staff are largely murdered and the device stolen to be used as a weapon by Khan, which wouldn't have happened if the USS Reliant didn't stumble across his Lost Colony while doing scut work for the Genesis Project.
Sacrificial Lion: Somewhat erratically so. Sometimes Kirk agonizes over his death, yet at other times he seems to shrug it off, or even forget him entirely. For example, in Star Trek: Generations, Kirk expresses shock that Sulu has a grown daughter, wondering when he had time to do so what with his career. This overlooks the fact that Kirk also had a grown child.
Sour Supporter: Is eager to express his displeasure with the situation, both before and after Khan makes his appearance.
Foil: To David Marcus. Both are younger men subordinate to an older man who acts as a parental figure to them, but while David is brash and wants nothing to do with Kirk (at first), Joachim is reserved and has Undying Loyalty to Khan.
While the film leaves it ambiguous whether Joachim is meant to be Khan's son or not, the Expanded Universe explains he's his adopted son, raised by Khan after his parents (his father being Joaquin from "Space Seed") were killed in battle.
Also a foil to Saavik. A younger crewmember who tries to be the voice of reason to their overconfident leader, only to be ignored with disastrous results. The key difference is that Kirk first ignores and then listens to Saavik, while Khan first listens to and then later ignores Joachim.
Only Sane Man: Granted, he's the only speaking member of Khan's crew so we never really find out what his other followers think, but he's certainly the only one who actually stands up to Khan.
Mook Lieutenant: Not really physically intimidating to be The Dragon, as Khan does all his own interrogations. He just exists to allow Khan to awesomely monologue to someone.
Morality Chain: Joachim tries to be this to Khan, but it doesn't really take.