As a biopic, it's accepted that the picture stays fairly close to the book's depiction of events. There is a slate.com breakdown that goes through several scenes and compares them between the book and the movie.
Where the accuracy is called into question is not in the book vs the movie, but in the book vs real life. The slate article notes a few of these, such as the following examples:
According to Thorson, Liberace’s mother Frances (Debbie Reynolds) pushed him hard to practice the piano as a child, forsaking playtime with other kids—and the film presents a similar story. These accounts differ, though, from Darden Asbury Pyron’s 2000 biography, Liberace: An American Boy, which suggests that as a child, the future star was more than happy to practice on his own (“His parents could not tear him away from the upright in his parlor”) and didn’t seem to mind not having friends.
In Soderbergh’s account, he moves out of Liberace’s mansion the day Thorson arrives; the memoir says that that Jerry stayed, despite everyone’s discomfort, for four more months.
This is further corroborated by other accounts such as the one from The Guardian:
This applies especially to his jaw-dropping assertion that the cosmetic surgery he underwent was forced on him by his famous lover, specifically to make him look like a younger Liberace.
There is no independent corroboration for this claim, but its unreliability only adds to the story's fascination: a bizarre, anti-Pinocchio parable in which the power of loneliness and toxic love transforms a handsome young guy into a deeply unhappy plump-nosed, cleft-chin latex doll.
Other criticisms include that it is only a short part of Liberace's life (i.e. the part where Thorsen is a part of it), and that the author may skew things either intentionally or unintentionally.
So in short, yes, the movie is fairly true to the book, but the accuracy of the book is definitely in question.