Biography
Early years
We know very little about the man currently using the name Jack Harkness (or Captain Jack Harkness or simply Captain Jack.)
Early in his life, he persuaded a friend to "join up" to fight against unspecified enemies, described by Jack as horrible. The enemies tortured his friend, so that Jack bore the guilt of his friend's fate. (TW: Captain Jack Harkness).
The future Captain Jack Harkness worked as a Time Agent, until he discovered that the Agency had erased two years of his memory. (DW:The Empty Child)
He would, at some point, acquired a sleek but small Chula spacecraft, fitted in for human use.
Basing himself out of the World War II time zone, he assumed the alias of Captain Jack Harkness, who died in action on January, 1940, though he knew very little about the "real" Jack, other than the latter's date of death. (TW: Captain Jack Harkness)
Jack worked as a con man, finding pieces of space junk and directing them to soon-to-be disaster sites, selling them to passers by, and letting them get destroyed before the buyers could pick up their merchandise.
Meeting the Doctor
While pulling a con with a Chula Ambulance during the London Blitz, he spotted Rose Tyler hanging from a barrage balloon. He rescued Rose, who introduced him to Ninth Doctor. (DW: The Empty Child). Together, they worked to stop the Empty Child plague brought about by the Chula Ambulance's nano-genes. Just before the destruction of his ship, with him in it., carrying as it did a German bomb about to exploded, he was rescued by the Doctor and Rose just before his original ship exploded. (DW: The Doctor Dances) and accompanied them on their subsequent adventures.
Jack was aboard Satellite 5 when the Daleks launched their assault on Earth, and was exterminated while defending the satellite from their advance. He was resurrected by Rose Tyler, who at that time had the powers of the Bad Wolf entity. Returned to life, Jack found he had been rendered immortal, although he was unable to rejoin the Doctor and Rose before the TARDIS departed, and was left stranded on the satellite. (DW: The Parting of the Ways) The Doctor, who has just regenerated, believed that Jack could begin the process of "rebuilding the Earth".
Torchwood
Somehow, Jack escaped the satellite and found himself on Earth in an unknown time period. After his ressurection, Jack could no longer died. He ultimately joined the Torchwood Institute. In the 2000s, he became the leader of Torchwood 3, based under the Cardiff rift. TW Jack held on to the hope of re-establishing contact with the Doctor, who he believed could help him. He revealed very little of himself to his team, even when pursuing a flirtatious relaionship with Gwen Coooper (TW: The Ghost Machine) and a sexual relationship with Ianto Jones. (TW)
Transported back to 1940 by the Cardiff rift, Jack met his namesake, the original Jack Harkness, and briefly romanced him, before returning to the 21st century. (TW: Captain Jack Harkness).
After the rift was finally opened by Owen Harper, Jack was forced to confront Abbadon, released from the rift by Bilis Manger. The beast was destroyed while attempting to leech Jack's life, though the exertion rendered Jack dead for three days, his immortality apparently unable to save him. He was brought back to life by a kiss from Gwen.
Shortly after his resurrection, Jack noticed the Doctor's hand begin to glow within its hyperbaric chamber, and witnessed the materialization of the TARDIS in the Hub. The Torchwood team arrived to find the Hub empty, Jack presumably having left with the Doctor. (TW: End of Days.)
Behind the Scenes
Jack Harkness' first name was originally "Jax".