There are very few comic elements in Shakespeare's Julius Caesar. There's no scene of comic relief in Julius Caesar that corresponds to the "porter scene" in Macbeth, or the "gravedigger scene" in Hamlet. Shakespeare seems to have been content to include a few puns, some well-placed ironic remarks, and a joke about poets, and let the humorous element of this play rest at that.
The play opens with two tribunes, Flavius and Marullus, walking through the streets of Rome. They encounter a carpenter, and a cobbler.
MARULLUS. But what trade art thou? Answer me directly.
COBBLER. A trade, sir, that, I hope, I may use with a safe con-
science, which is indeed, sir, a mender of bad soles. ...
FLAVIUS. Thou art a cobbler, art thou?
COBBLER. ... I am indeed, sir, a surgeon to old shoes; when they are
in great danger, I recover them. (1.1.12-14, 24-26)
On his way to the Senate, Caesar stops to make a condescending remark to the soothsayer who warned him, "Beware the ides of March!" when Caesar first appeared in the play (1.2.21).
CAESAR. (To the Soothsayer) The ides of March are come.
SOOTHSAYER. Ay, Caesar; but not gone. (3.1.1-2)
The day's not over yet, Caesar.
Following Caesar's triumphal appearance before the people of Rome, Cassio wants to know what Cicero thought of Caesar's response to being offered a crown.
CASSIUS. Did Cicero say anything?
CASCA. Ay, he spoke Greek.
CASSIUS. To what effect?
CASCA. ... [T]hose that understood him smiled at one
another and shook their heads; but for mine own part, it
was Greek to me. (1.2.281-2870
There's a scene with a few citizens of Rome and Cinna the poet—not to be confused with Cinna, one of Caesar's assassins—that might suffice as a scene of comic relief, particularly since it occurs shortly after Caesar's assassination and directly after the funeral orations by Brutus and Antony.
Citizens who have been incited to rebellion by Antony's oration for Caesar confront Cinna and comically pepper him with questions while giving him no time to respond. They repeat the questions, one by one. Finally, they ask Cinna his name.
THIRD CITIZEN. Your name, sir, truly.
CINNA THE POET. Truly, my name is Cinna.
FIRST CITIZEN. Tear him to pieces; he's a conspirator.
CINNA THE POET. I am Cinna the poet, I am Cinna the poet.
FOURTH CITIZEN. Tear him for his bad verses, tear him for his bad verses. (3.3.25-30)
This scene might also be a satirical reference by Shakespeare to the ease with which a mob mentality can be engendered in a population, even though at Shakespeare's time, the Earl of Essex failed miserably trying to raise a rebellion against Queen Elizabeth I—and paid for it with his head. The citizens who confront Cinna are so mindless that they can't distinguish between a conspirator to a political uprising and bad poetry.
It could be argued that Caesar himself is a comic, buffoonish figure. Shakespeare's Caesar is boastful, self-absorbed, foolish, easily misled, and oblivious to the world around him—the world which he supposedly rules.
Caesar considers himself an astute judge of character.
CAESAR. Yond Cassius has a lean and hungry look;
He thinks too much; such men are dangerous. (1.2.201-202)
Caesar is right about Cassius, of course, but he fails to discern the true character and political motivations of those closest to him, including Antony and Brutus.
They stab Caesar.
CAESAR. Et tu, Brute? (3.1.85)