This is a mediocre Hammer film, directed by Terence Fisher (Horror of Dracula). Christopher Lee is the biggest name in the cast, but he isn't the lead ... he's not even the villain. That job goes to Anton Diffring (Circus of Horrors), who was in everything back in the day. Perhaps the most notable member of the cast is Hazel Court, one of the first Scream Queens, and an early example of the kind of actress Hammer liked to put in their movies (i.e., cleavage). Court says they even filmed a brief nude scene "for the European market" ... she's modeling for a sculptor. No copies of that version exist. All we have is one still of a topless Court, which was a big deal in 1959. (In fairness, Court in the nude is still a big deal.) The censors may have succeeded in burying the nudity, but throughout the movie we see a bust of the sculptor's work, which shows Court's assets off quite clearly.
The movie is almost all talk, and thus, almost entirely dreary. The plot is a mishmash of mad scientist and Dorian Gray. None of it is particularly interesting. Among the films it played with in the theaters: The Evil That Is Eve (aka A Kiss for a Killer) and Dr. Terror's House of Horrors. The Man Who Could Cheat Death will cheat you out of 83 minutes of your life, time that is better spent using Google to find that still of Hazel Court (you'll still have 80 minutes left over to do what you want).