I'm not sure that there is a climax in the book. It is a biography of Henrietta Lacks, so it deals with her life, her family and the contributions her cells have made to science. The book is actually as much about Henrietta Lacks as it is Rebecca Skloot's pursuit of information on Henrietta. If you want to consider a turning point, I would suggest the moments when Skloot uncovers more of Henrietta's life and certainly the moments when Henrietta's impact on cancer research becomes more publicly known and acknowledged. One such moment occurs in Chapter 28, when a medical conference was held in Henrietta's honor and the Lackses were invited. In addition, the BBC began filming a documentary about her.
In Chapter 32, Rebecca accompanies Zakarryia and Deborah to Johns Hopkins where they actually witness one of Henrietta's cells dividing. They are surprised to hear Christopher Lengauer admit that Johns Hopkins "screwed up" by never communicating with the Lacks family. Uncharacteristically, Zakarryia is silent on the way home, prompting Deborah to tell Rebecca that she just witnessed a miracle. This is a defining moment in the biography because they actually see Henrietta's cells and they get some acknowledgment from Johns Hopkins that they had not properly handled the acknowledgement of Henrietta's and her family's contribution.