Vandal Jax does not appear to be related to the gang that has surfaced among the other champions with the vandal skins such as Twitch, Brand, Vladimir and Gragas. This is further supported by its release date, which preceded the other Vandal skins by a significant amount.
Jaximus is a reference to the Murmillo, a gladiator class from the Roman Imperial Age.
It might also be a reference to Maximus, the protagonist of the movie 'Gladiator', although Maximus wields a sword, rather than a polearm.
This skin features a memorial tribute to Joe, a Ewings Sarcoma patient who visited Riot thanks to the Make A Wish Foundation. Joe passed away on July 13, 2012. The movement quote for the Jaxiumus skin, Joe's favorite, was changed to "Here's to you, kid!" in his memory.
PAX Jax is a reference to the Cardboard Tube Samurai from the web comic Penny Arcade (which hosts the Penny Arcade Expo - PAX). A symbol on the skin resembles Pac-Man and matches symbols on the Samurai's gi and the shirt of Penny Arcade protagonist Gabe.