There are numerous similarities between Lord of the Flies and Julius Caesar. A comparison can be made between the character of Ralph from Lord of the Flies and Marc Antony from Julius Caesar. Both characters play a major role in the storyline as well as the theme. They are responsible for pointing out the faults in the other characters to the audience. In both stories, Ralph and Marc Antony try to do what is best for the other characters of the story. Unfortunately, the motives change as the story develops and their actions become more self-centered. A second comparison can be made between the characters of Simon from Lord of the Flies and Julius Caesar from Julius Caesar. It was through the tragic death of the characters that the breakdown of the moral fibers of the story was revealed. Both characters were “silent characters” but were responsible for offering crucial information about the other characters. A final comparison can be made between the imagery found in both novels. In particular, the graphic depiction of Roger sodomizing the pig in comparison to the heinous murder of Caesar (stabbed 32 times). It is through the use of imagery that Golding develops the idea of man corrupting the island (theme) and Shakespeare develops the idea of ambition being the corruption of man (theme). Hopefully this helps.
What about comparing the gang mentality in each work? In Lord of the Flies, Jack is able to go against Ralph, even though Ralph was chosen as the leader, and get his own group of "hunters" together that eventually turn into savages that wind up killing Simon and Piggy. In Julius Caesar, Cassius and his fellow conspirators are able to convince the other senators, including Brutus (who is Caesar's friend) that Caesar is too ambitious and should be killed before he turns Rome into a dictatorship instead of a democracy. Cassius wants to kill Caesar because he is personally envious of Caesar. Jack is envious of Ralph.
You could explore the themes of good vs evil in both works. As Cassius says, "The fault dear Brutus is not in our stars but in ourselves." This means that the evil is within them, not in the stars or because of the gods. In Lord of the Flies, the Beast is within the boys, it is their inner evil, not a result of society. Another theme is reason vs emotion. You could talk about how Cassius uses reason to convince even Caesar's good friend, Brutus, to join the conspiracy, just as Jack is pretty convincing to get the boys to become hunters and forsake Ralph's leadership. When the boys kill, their emotions are way out of control -- they are in a frenzy. Or, you could compare Jack as a character to Cassius. They are both villains.
Read about the themes of both works here on enotes and you can get some other ideas for comparison.