The only real "prop," i.e., property, is the cloak. Shakespeare could not show Caesar's bloody body--or at least he chose not to do so. Instead, he followed Plutarch in having Antony arouse the crowd by exhibiting Caesar's bloody cloak. In order to do this, Shakespeare must have had two cloaks that were exactly the same except that one was in good condition and the other was shredded and bloodied with imitation blood or with real animal blood. Caesar puts on one of these props when he decides to go to the Senate house, against his wife's urgent pleas. Then when Antony is making his famous funeral address he shows the other prop to the crowd as evidence of what the conspirators have done to poor Caesar.
In Act II Scene II, after Calpurnia has failed to persuade her husband to stay at home, Caesar says:
How foolish do your fears seem now, Calpurnia!
I am ashamed I did yield to them.
Give me my robe, for I will go.
The scene does not end with these fateful words (which in Hollywood is called a punch tagline) but is followed by some inconsequential dialogue which is obviously intended to allow time for a servant to bring Caesar's robe and help him into it. The audience is given plenty of time to observe this robe, so that they will think they recognize it when Antony holds it up, all torn and bloody, for the plebians to see. No doubt Caesar's two robes were of a distinctive color so that they would be easily recognized. As a matter of fact, Antony tells the mob that the first time Caesar put it on was on a summer evening. This might be intended to explain why the robe was of a distinctive color. It might be a light-colored robe to contrast to the dark robes that all the other actors are wearing at the Ides of March, when it is still not quite spring yet. It the two props, the two robes, were white, or nearly white, the ruined one would show up the bloody gashes most effectively. These two props would become permanent properties of Shakespeare's company to be used whenever Julius Caesar was staged.
Caesar's body was probably never shown on the stage. It is supposedly inside a coffin covered with the bloody robe. When Antony holds up the robe, the mob looks into the coffin and pretends to react to the sight of the ravaged body--but the audience does not see the body, only the robe. The audience does not see the body because (1) the body is inside a coffin, (2) the mob is blocking the view of the coffin, (3) Antony is shielding the body with the robe, and (4) Antony is distracting attention with the horrible vision of the robe. It might be that the actor playing Antony held the robe up in such a way that it completely hid the audience's view of the coffin.
When the stage directions say that Antony enters carrying Caesar's body, what is actually happening is that Antony enters carrying a dummy completely covered in the second prop, the one that has been shredded as if by assassins' daggers. Shakespeare must have thought that he could arouse more emotion in his audience with the bloody robe than if he tried to show Caesar himself covered with fake wounds and smeared with animal blood.
It would also be difficult for an actor to carry another actor onto the stage. (King Lear does this with Cordelia in the last act of that play, but "she" is surely much lighter than the actor playing Julius Caesar.) Such actions are very hard to stage. The actor playing Caesar would have to be pretty light, and the actor playing Antony would have to be pretty strong to bring this off without a lot of grunting and groaning. Marlon Brando carries the body of Caesar in the motion picture version of Shakespeare's play in which Brando played Antony, but he was an exceptionally powerful man and was able to carry another man who must have weighed close to two hundred pounds without any apparent effort. What they do in movie versions of Shakespeare's plays can be quite different from what Shakespeare was able to do himself.
In the "friends, Romans, countrymen" funeral oration, the main props that Marc Antony uses are Caesar's will, his body and his cloak.
Antony's intent for all of these (an intent that is fulfilled) is to stir up the people of Rome against the men who assassinated Caesar.
He uses the will to show what Caesar had left to the people of Rome -- that he has left money to all of them.
He uses the cloak and the body to focus their attention upon the brutality that was done to Caesar -- while he does this, he describes all the details of the killing.
Each of these things works to incite the crowd to a frenzy, which is ironic, because they had been just as excited by what Brutus had said just a little while before in justifying the killing of Caesar.
Caesar's will, body and cloak are used to evoke sentiments amongst the plebians.