How can I portray a character who does not actively seek attention, but is still vain and self-superior?
Since I don't know how much information you will need, I will list everything I think might be relevant below, so you can choose what you think matters to read:
Some monsters attack Earth, the protagonist boy discovers his fancy new power and is taken by a bunch of good aliens. Good aliens tell the boy that there are bad aliens that want to attack Earth, and they must train the boy to defeat the bad aliens. The character in question is one of the good aliens
Character in question is not the protagonist, but the most important non-protagonist character, if not more important than the protagonist himself.
This character is named Alice, she is female with no stereotypically feminine traits. She is 17-20 years old. Her parents are the good alien's leaders. She was raised as an experimental super-soldier in preparation for the war with the bad alien.
Oh, and even though Alice is an alien specimen, consider her to be identical to be a normal human being only raised in a different society, because the good aliens' bodies and psychology has no important difference from that of human beings.
Tone and atmosphere
The atmosphere is pessimistic and grim-dark-ish, because the bad aliens are unimaginably strong and seemingly infinitely numerous, while the good aliens are weak and have been hunted for generations, with one crushing defeat over another. The good aliens' society is not what most modern readers would consider ideal either—they are utilitarian and fascist, like District 13 from the Hunger Games trilogy.
As to the Earthlings, they are just helplessly struggling in chaos and have no significance to the story beside something that the protagonist thinks he's trying to protect.
More about Alice
Alice herself is pessimistic, selfish and cynical, with no desire to help her species nor empathy for anyone specifically; however, being an experimental super-soldier and one of the expected inheritors of leadership, she is also the focal point of the good aliens' remaining hope.
She enjoys this kind of attention, but find the people paying her such notice to be foolish and merely amusing. With such an attitude and social position, she believe herself to be superior in intellect and power, hence the arrogance.
Role in book and plot
Alice is the character foil of the protagonist, or vise versa—where the protagonist boy (currently unnamed but may be referred to as "Bob") is innocent, superficially altruistic and acting upon emotional impulses. Alice is cynical, calculatingly selfish and acts only so as to advance her own welfare.
When the good aliens adopt Bob as a tool to deploy against the bad aliens, Alice feels her previous prestige and attention were taken away, and is thus somewhat spiteful towards Bob. She is also aware, with some of this awareness extending beyond the fourth wall, that the arrival of Bob into the midst of the good aliens' society is like the arrive of Neo amongst Morpheus's team, like it is in The Matrix, which implies that she will inevitably become a support character like Trinity, something she strongly despises. Yet, being the most successful of the super-soldier program, she must train alongside Bob and is obliged to sometimes offer guidance.
Alice will eventually enter a redemption arc, but that is after it is necessary to portray her as a passive yet arrogant person.