I agree with Roger Ebert's insightful review which motivated me to watch the movie, I highly recommend it. I will add some points.
DiCaprio's performance is brilliant, very through. It reminded me the character Citizen Kane in the classic film of the same name, directed and starring by Orson Wells seventy years ago, considered by many the best American film, nominated and winner of Academy Awards. Because of the issue of citizenship, the degree of conviction of both characters, and how was depicted by bodily mechanical movements that express their ability to have a cold mind. DiCaprio manages to represent his physical contundency, even though his height is lower than that of Orson's, though had gained some weight for this purpose. I would not be surprised to know that DiCaprio has carefully studied Orson's performance for his characterization of Hoover.
I have two problems with the movie. Clyde's character looks good to me until old age begins as his makeup is not convincing, not well done. I am surprised that Clint Eastwood, being such an excellent director, had not demanded better. In contrast, Helen's and DiCaprio's are very good, evident from the first scene in which Hoover appears mature. Also, Clyde's performance is weak specially in the scene when he finds Hoover dead on the floor, suffering from superfluous sentimentality, affecting the movie's balance.
The most interesting relationship is between Hoover and Helen. He seems to respect her. He expects from her, of course -as of others- loyalty, but not surrender. It is to her that he asks what defect he has for her not to accept his "proposal"?. She replies: "I'm not interested in marriage. The most important thing for me is my work." Or he says: "Why do I kill everything I love?" Helen's loyalty is disinterested, based on her passion for her work; it is a logical consequence of its nature. We see in the end that she destroys the secret files, complying with her oath to Hoover's petition, which seems to imply that he was only keeping the secret files to ensure his position in power, for serving and saving the country, and did not have the ulterior motive of destroying public figures per se.
The film is an excellent reflection on masculinity, in those days and these. What is a man? A hero? DiCaprio is in a very comfortable position for the Oscar as best actor. Actress Naomi Watts also is an excellent candidate for the Oscar as best supporting actress. Even the movie title is ingenious as it is the proper name, J. Edgar, misleadingly insinuating greater intimacy, closeness to Edgar, while Eastwood retains the privacy of the character as far as he understands Hoover's masculinity permits.