You mean the book / movie about Michelangelo and Pope Julius?
I've never read the book (although from a couple of reviews / synopses it looks to be fairly biographical & well researched), but the movie is... well, it takes some artistic licenses.
Here are some of the facts and instances that the movie got right:
Michelangelo was occupied with sculptures for the Pope's tomb when Julius commissioned him to start work on the ceiling frescoes. He was reluctant to do it because he considered painting a 'lower' art form, and was primarily (and preferably) a sculptor.
Julius originally wanted a cycle of the Twelve Apostles as the decoration; Michelangelo negotiated for a grander scheme of his own design (although how much of it was truly his own design is still under academic dispute).
Julius and Michelangelo did have a rather fraught and tense relationship, which manifested in various disputes over the imagery contained in the frescoes, Michelangelo's work environment, and the time he was taking to complete the commission, amongst others (Julius had a small window at the top of the altar wall - where The Last Judgement would eventually be painted - to watch Michelangelo and ensure that he was working).
And a few examples of where it went wrong:
Michelangelo never begun work on the roof before fleeing Rome due to the war (and so he could continue sculpting). He signed the contract to do the ceiling after returning to Rome. He did go to Carrara for marble for the tomb, but before Julius' initial request for the Sistine frescoes.
He didn't convince Julius to change the design from the Apostles while on a battlefield.
The commission was never given over to Raphael. Raphael did work on portions of the Vatican (most famously the four 'Raphael Rooms', including the School of Athens in the Stanza della Segnatura), and Vasari stated that Raphael was given access to the Sistine Chapel while it was being painted, but he used this as inspiration for his own works.
Contessina de'Medici was a real person, one of Lorenzo de'Medici's children, but she was not Michelangelo's lover.
If you're interested in Michelangelo and the Sistine Chapel, there's a ton of literature available (for fairly obvious reasons, it's a very popular topic). I recommend Massimo Giacometti's The Sistine Chapel and Anthony Hughes' Michelangelo as a couple of notable texts amongst the mass.