The greatest charge against Caesar, as laid out by Brutus, is that Caesar put his personal ambition before the state's well-being. Brutus believes this to be a strong enough charge against Caesar that all Rome will agree with him once he points it out. Of course, Antony levies the same charge against the conspirators and eggs on the crowd to the point that they call the companions of Cassius and Brutus traitors. As the crowd gets ready to riot, it now seems that the conspirators acted against the interests of Rome, as it was embodied by Caesar.
Loyalty's central importance is best brought out by the exchange between Brutus and Cassius versus Antony and Octavius before the battle. Octavius declares he won't put away his sword until Caesar is avenged, or until he, Octavius, dies on the swords of traitors. Brutus points out that Octavius can't be killed by a traitor unless he brought one with him, since Brutus and his men are no traitors. Octavius challenges him again, saying he wasn't meant to die on Brutus's sword. Brutus responds that, actually, he couldn't hope for a more honorable death than that, implying that Brutus only ever kills out of honorable intentions (like, for instance when he killed Caesar out of his loyalty to the state).
They go into the battle, and we know how it ends, but Brutus's death and his call out to Caesar are made all the more poignant by his characterization of himself at the pivotal moment of the play, when he decided he had to kill Caesar. Though he loved the man, his highest duty was to Rome. This loyalty to his state justifies his murder of Caesar, but in the end, it comes to justify Brutus's own death. He would not be brought back in chains to Rome, as it would be a disgrace to the city. Instead, he honors his country by doing himself in, because, as he said, "I love the name of honor more than I fear death."
Ancient Rome was all about rhetoric, the art of speaking persuasively. Antony claims that if he were a better speaker, he'd be able to sway the crowd to mutiny. Of course the power of rhetoric is manipulating people without them realizing they're being manipulated. Cassius tries to do this when he suggests to Brutus that he's as good as Caesar (hinting that Brutus should rule instead of Caesar). Caesar is also his own undoing – while the men are kneeling to finish him off, he gets caught up in his own transparent rhetoric. Caesar means to compare himself to the North Star to show how firm and constant he is in his opinion, but his rhetoric betrays his arrogance, showing that he holds himself up above other men.
The most powerful rhetoric in the play by far is Antony's speech to the crowd. It's masterful for its use of irony – literally, saying the opposite of what you want your audience to hear. Antony's speech works because he doesn't come out and say Brutus and his buddies are traitors. Instead, he suggests it, which means the crowd thinks they come to the conclusion on their own. You're always more likely to believe your own ideas, even if they've been planted by someone else.
So eulogies are those big speeches in praise of a dead person, particularly at their funeral. The word is related to elegy, which was originally a poetic form often used to deliver poems of mourning. Unlike in a lot of Shakespeare's other plays, the eulogies in Julius Caesar don't really have mourning at their center – they appear to, but there's actually some other goal achieved. When Portia and Cassius die, Brutus doesn't make great speeches of mourning; he honors them quietly. By contrast, Antony is the source of the play's two main eulogies, and if we know one thing about Antony, it's that he's slippery.
Antony specifically plays on the notion of the eulogy when he begins his speech at Caesar's funeral with, "I have come to bury Caesar, not to praise him." Of course, he does praise Caesar (again with irony), and the eulogy stops being an occasion for mourning and becomes instead a call to action.
The play's other notable eulogy is delivered by Antony about Brutus. It's strange that Antony, who specifically set out to get Brutus banished and killed, should be the one to call him "the noblest Roman of them all" – especially given that, like ten minutes ago, he was calling Caesar the noblest. This could be us being cynical, but when you take the two eulogies together, Antony doesn't come across as very sincere or trustworthy. What's more, we know that he will go on to betray Octavius (after the events of the play).
Though Antony's words for Brutus are nice and all, Brutus really gives us his own eulogy when he commits that incredibly noble act of self-sacrifice: taking revenge against himself to avenge the death of his friend Caesar. "Caesar, now be still, I kill'd not thee with half so good a will" is all the eulogy Brutus needs, because it tells us how, why, and for what he died.