Not much is known about Jack. Thus far, Jack has only been mentioned three times.
- In chapter 692 during the failed experiment by Caesar
- In chapter 801 during the escort of Doflamingo
- In chapter 805 during the explanation of the destruction of Zou.
The only thing we seem to learn from this is that he is related to Kaidou, but we don't know the reason he was send by Kaidou to Doflamingo and why he destroyed Zou. My personal guess would be that Kaidou sent Jack to Doflamingo (as was seen in number 1) to prepare for the last battle as Kaidou said in chapter 795:
Goddamn it, hurry the hell up, Joker! Get everything ready for our last battle! A world this mundane isn't worth keeping around! Time to light the match to the greatest war this world has ever seen!
Jack probably came by Zou and destroyed it, because that seems to be what he does. Kaidou's underling seemed to have warned Kaidou something bad would happen if they send out Jack.
Finally, he isn't one of the 7 warlords. The five remaining confirmed warlords are Mihawk, Kuma, Hancock, Buggy and Weevil after Doflamingo and Water Law were stripped of their title. There has been no mention of a new warlord ever since Fujitora apologized for such a system even being in place.
My personal guess would be that there might not even be more warlords. It seems many of the Marines dislike the system (eg: Sengoku, Fujitora, Smoker, etc) and they might want to take this opportunity to get rid of it. Especially seeing 3 out of 6 ex-warlords blatantly abused the system to get away with their evil deeds and did more harm then good by being a warlord (ie: Crocodile, Doflamingo and Teach)