Unfortunately, you are out of luck. There is no published biography of John Rackham (by the way, his "Calico" nickname is also something probably made up by Charles Johnson), probably since his career by himself was not long nor notable. He sailed as quartermaster for Vane for a while and ended his career in a short two-month voyage with two women who are now more famous than him.

If an academic did want to create a history of just Rackham's career, they would be hard pressed to get past something that was journal article in length. Outside of looking into Vane's career and his final voyage, the very brief career between leaving Vane and the Bonny-Read voyage of two months has, as far I know, there are only three primary sources to describe it with short references to what happened. It's probably why Johnson's account of him is so short in that period (and while I haven't looked at it in a while, I suspect it's not far from matching what we know from the period documents). It's also probably a reason why he could only rally a tiny crew for his last voyage and had to bring two women with him.

Pretty much, I think you will be out of luck, since there isn't terribly much to talk about for his career. Though, now that your question made me think about it, I will probably add to my list of articles to do for my site. You might be interested in my article about Anne Bonny, Mary Read, and women in the maritime/pirate world of that time.

Any reason why you're looking? If it's anything to do with Black Sails, I'm sorry to say that the historical record does not resemble that show's depiction much at all, or at least the historical record does not suggest that Rackham was like the one in the show.