Corakinus described his kind as "an accident", saying they should never have been born into a universe of light. (TV: Co-Owner of a Lonely Heart) As April MacLean explained, the Shadow Kin believed that they were an evolutionary mistake, "like a peek into the universe's dark side that it never wanted", and so they were "cast down [into the Underneath] to live in [the universe's] shadows and be forgotten."
This was a major drive behind their invasions. They also believed that the universe would crush them if they didn't defeat it. April called this a part of their religion. (TV: Co-Owner of a Lonely Heart)
The Shadow Kin could exist as "pure shadow", living unnoticed and untouchable in other people's shadows. In that form, they could only be killed if the person casting this shadow was killed as well. They could take on a solid form as well to kill their victims by surprise. Even in their most solid form, they could not exist when there were no shadows. (TV: For Tonight We Might Die)
The Shadow Kin emerged on the planet Rhodia amidst a war between the Rhodians and the Quill. Taking the native population of three billion off guard, the Shadow Kin drove them to near extinction with the exception of the Rhodian prince Charlie and a single Quill forced to serve as his protector as a punishment, who were relocated to Earth by the Twelfth Doctor.
Tracking the fugitives to Coal Hill Academy, the Shadow Kin, led by Corakinus, emerged through a tear in space-time at the school's prom night where they proceeded to mercilessly hunt for the pair before they were ultimately confronted by the Doctor, who Miss Quill had summoned for help. When the Doctor used the Shadow Kin's aversion to light against them, Corakinus and his kind were forced back through the tear from where they came which the Doctor subsequently sealed, though not entirely. Corakinus warned, however, that the Shadow Kin would return. (TV: For Tonight We Might Die)
After Kharrus helped create a link to April on Earth using an anchor of a life blood, Corakinus planned to send his army to attack Earth. (TV: Co-Owner of a Lonely Heart) When April found her way to the Shadow Palace in the Underneath and defeated Corakinus in battle, she became the new king, and had the Shadow Kin lock up Corakinus. She ordered them to kill the killer petals in Shoreditch on Earth, and had them close the crack that brought them through on the way back to the Underneath. (TV: Brave-ish Heart)
They later returned back under Corakinus' command, and they use other smaller tears to enter back to Earth. Corakinus revealed his intent to take April back to the Underneath with him, so he could have her heart in his body forever. When Charlie was posed with an tough decision, April read Corakinus' thoughts and realised that the Shadow Kin were planning to wipe the human race out regardless of whether she went with them or not. Charlie decided to use a displacement gun to kill April and Corakinus, thus allowing Charlie to become the new King. He ordered his army to retreat, and then he used the Cabinet of Souls to wipe them out. After he used the cabinet, April resurrected but in Corakinus' body. (TV: The Lost)