The Jackson is the son of Jack-in-the-Box from a possible future timeline in which The Wasting occurred. Raised by the Brothers of Trouble, he is a bloody-minded fanatic, perfectly willing to kill in his pursuit of criminals. Obsessively devoted to a religion revolving around his father, he has even surgically altered his body to be the ultimate fighting machine.
In an alternate reality in which Zachary Johnson never handed off the Jack-in-the-Box role, he perished in the line of duty shortly after the birth of his son, the child who would one day become the Jackson.
In this timeline, The Wasting occurred, but the infant was delivered from it by time-traveling Brothers of Trouble. They took him into their future, where they raised and indoctrinated him in their faith. Growing up, he devoutly studied his father's exploits and quotes, which in grossly distorted form served as the basis of his religion, in preparation for one day carrying on the Jack-in-the-Box legacy. He even underwent extensive, cutting edge body modification, tailoring his very flesh for the role.
The Jackson eventually traveled back in time to meet his father. In his past (our present), he stopped The Box, an alternate version of himself visiting from a different future, from murdering Jack-in-the-Box. However, he soon decided that his fabled father was too soft on crime, a traitor to his own tenets (as the Jackson perceived them), and allied with Box to attempt patricide. ("Serpent's Teeth")
While in police captivity, he was pulled back to his native timeline. ("Father's Day")
Extensive body modification has augmented the Jackson beyond human limits, into a deadly, animalistic hunter. For example, he can leap great heights and distances, mimicking his father's own footapult-aided jumps. His hands end in deadly claws, and he has an enhanced sense of smell. Additionally, the Jackson's veins are capable of secreting an acid powerful enough to burn through Jack-in-the-Box's confetti.