This is Deviants (2012), the first book of the Dust Chronicles trilogy by Maureen McGowan.
In Glory's world, different means deviant—and dead.
In a post-apocalyptic world, where the earth is buried by asteroid dust that’s mutated the DNA of some humans, an orphaned sixteen-year-old, Glory, must hide and protect her younger brother. If their Deviant abilities are discovered, they’ll be expunged—kicked out of the dome to be tortured and killed by the Shredders. Glory would give anything to get rid of her unique ability to kill with her emotions, especially when Cal, the boy she’s always liked, becomes a spy for the authorities. But when her brother is discovered, and she learns their father, who was expunged for killing their mother, is still alive, she must escape the domed city that’s been her entire world.
Outside in the ruins, they’re pursued by the authorities and by sadistic, scab-covered Shredders who are addicted to the lethal-to-humans dust now covering the planet. Glory’s quests to transport herself and her brother to safety make up the thrilling and fascinating first volume of The Dust Chronicles.
I think the sister can use her eyes to kill and control.
In the first chapter, Glory hunts rats to feed her brother. The hunting process is meeting the rat's gaze, and shaping her emotions into killing it.
One skitters into a sliver of light and lifts its head to make eye contact.
Big mistake, Mr. Rat.
Held in my gaze, the rodent can’t look away. Emotions heighten my senses, and soon I can feel the rat’s rapidly beating heart, hear its blood coursing as adrenaline floods its veins. It’s as if my fingers are pressed to its pulse, my ear to its chest. But they’re not. The sensations build until the rat’s completely under my control. [...]
Focusing my power, I picture the rat’s heart, sense it compressing, and will my emotions to squeeze. [...]
The rodent seizes, every muscle stiffening at once. Its heart rate slows, then it gasps and falls on its side, legs twitching in death throes. Sympathy creeps up my throat, but I push it back down; one rat won’t fill Drake’s belly for long. When I’m sure it’s dead, I pull our dinner forward by the tail and find another victim—then another.
The brother can harden his skin.
There's an indication of sorts later in the same chapter:
But I’m luckier than most. At least my curse is easy to hide. When my brother’s hits, his skin changes...
They both live in this dome that protects them from the outside, which has dust that can kill them.
The dust certainly kills humans, but it's also implied that it kills cattle, since the city has to breed rats. Rats are said to be the only species able to survive the dust. The dome protects the citizens both from said dust, but also monsters called Shredders.
I gesture toward the fugitive rats that must have escaped from a farm factory, where rats are raised and slaughtered for food. Or they may have breached the dome from Outside. Out there rats are the only animals that can survive the dust. Rats and Shredders.
Found with the Google query
book kill eyes dust dome skin brother.