The CIA had nothing to do with his death. I've never heard that before.
Now, it has been rumored that the Vatican wanted him dead and poinsoned him. However, the theory that he was poinsoned is flawed in that it suggests he died at around 4:30 a.m., but his body was found at 5:30 a.m. and rigor mortis had already set in. The theory saying he was poinsoned requires several hours before he'd have died, which would have put his death at around 4:30 a.m.
John Cornwell investigated the reports of the Pope being poisoned, but came to conclude, based on the Pope's niece (who was a medical doctor, by the way), that he was already in poor health. Cornwell concluded that pulmonary embolism was the cause of death. This cause of death fits perfectly with the Pope's past medical history.
Also, the person who died a week prior was not poisoned (that was a conspiracy theory as well). He did die, but he was not poisoned, as wikipedia will explain.