Jax was born sept, 21, 2350, in Paris, Earth. He's the great-grandson of James T. Kirk, hence his middle name. His parents were Thomas Vincoer DeSabel and Carol Tiberius (Marcus) DeSabel (named after her grandmother and grandfather). Carol was the daughter of David Marcus and Saavik.
Jax grew up in Paris with his aunt and uncle, Colleen and Peter Allain, until age 11. Then he went to Vulcan for about two years. His parents, of course, were always off somewhere on their ship, the U.S.S. Curie, as they were both science officers in Starfleet. Still, Jax saw them as often as possible, over the years.
But when he was 13, his parents died in the line of duty, when their ship was destroyed by an unknown assailant. There was some discussion about what would happen after that, but ultimately it was decided he would accompany his godmother, Denise Kallerinth (also a cousin, or something) on her new assignment: she was an engineer in Starfleet, and had just been transferred to the U.S.S. Enterprise-D. Jax's godparents, Denise and her husband, Jaak Luukarn, had been Jax's parents' closest friends.
On the Enterprise, Jax met and befriended Sam Lator and Wes Crusher. Sam and Jax soon became best friends. (A friendly rivalry grew between Jax and Wes.) At age 15, Sam and Jax became acting ensigns, and were sent to late 20th century Earth to conduct atmospheric studies, which took them two years of their own time (tho they made frequent visits home to the 24th century, and the actual time they covered in the 20th century was closer to a decade). For this mission they created AI programs and small robotic bodies for them, though much of their time is spent outside these bodies, often being downloaded into different computer systems and the like. Jax called his AI Boulder and Sam called his Nuggit. Bole and Nug do not belong to Jax and Sam; they are legally sentient individuals who since the completion of the 20th century mission do whatever they like, tho they consider Jax and Sam among their best friends, and visit them frequently, occasionally also working with them.
In 2367, Sam and Jax entered Starfleet Academy. Jax majored in engineering and computer sciences, while Sam majored in science, with a minor in engineering. The work they did in the the 20th century was counted toward their class time, so they entered the Academy as juniors (on an accellerated program of study). At the Academy they met a number of new friends, including Alex Franx, a security major. The three of them graduated in 2369, and were sent to DS9. Jax worked under O'Brien, Alex worked under Eddington and Odo, and Sam worked under Dax and O'Brien.
On one of their missions, the three of them came into possession of a wrecked runabout. Starfleet was going to scuttle it, but the ensigns convinced them to give it to them. They fixed it up, making improvements on it beyond normal runabout capabilities. All standard features were enhanced, and new ones were added, including several they designed themselves (and some which others had designed, and the trio shouldn't have had, officially). Metaphasic shielding, Pegasus cloak, transwarp conduit generators, quantum torpedoes, etc. they called their runabout the Musketeer.
If you go much beyond this point in Jax's life, things begin to change depending on which universe/continuity you're reading about. There might also be a few differences earlier, at least in detail, but basically all this will be accurate for any universe this side of the mirror. And some of the things that happen beyond this point may also be similar.
At some point (I'll let you know when I figure out when), he bought the old Kirk farm in Iowa, which had been out of the family for awhile. It's run by foreman Kevin Judd.
Jax is an avid fan of entertainment of all sorts, especially from 20th century Earth. He has all sorts of movies, TV shows, music, books, comics, and stuff like that on file.
Sam, Jax, and Alex have an enhanced computer program they designed, which they call Mage.
Jax is at least rudimentarily skilled in various martial arts from Earth, Vulcan, Qo'noS, and perhaps a few others. He also speaks several languages fluently (and some that he can at least get by in, in casual conversation). They include Federation Standard, English, French, Irish Gaelic, Vulcan, Klingon, and some Bajoran and Romulan.
He has a few implants, some of which he designed or modified himself. These include a memory buffer chip which allows him to store data he may want or need to have on instant recall, as well as blocking (but not erasing) memories he may not want; it perhaps has a few other functions. He also has a subdermal transponder which is undetectable itself, and which can send a signal which is also undetectable to anyone without the proper subspace frequency & code. There may be some implants I'm forgetting, and he may add more in the future. But he won't let anyone- not even Q- get away with comparing him to a Borg.