What do the conspirators believe they have accomplished by killing Caesar?
The answer to this question lies specifically in the statements made by the conspirators soon after Caesar's assassination in Act 3, scene 1:
Liberty! Freedom! Tyranny is dead!
Run hence, proclaim, cry it about the streets.
Some to the common pulpits, and cry out
'Liberty, freedom, and enfranchisement!'
The two express the sentiments of all the conspirators. They all believed that killing Caesar would provide them freedom from a tyrant. They think all will be equal once they take power. They believe that they act for the greater good--the security and stability of Rome. This ideal is expressed in a variety of ways by Cassius and Brutus.
In their conversation in Act 1, scene 1, Cassius told Brutus about Caesar's infirmity and feebleness by describing a number of incidents in which the general showed weakness. He expressed amazement that such a fragile man should alone lead them when they had greater strength and ability to do so. Brutus also expressed doubts about Caesar's ability to lead and was fearful that the general might abuse his power if he should become emperor. He acknowledged that he would rather be a humble villager than call himself a citizen of Rome under the difficulty of Caesar's possible leadership.
He later repeats these sentiments in a monologue in Act 2, scene 1, after he had agreed to conspire with Cassius and others to ensure that Caesar does not become emperor. He has made up his mind that the general must die and provides reasons for this:
It must be by his death: and for my part,
I know no personal cause to spurn at him,
But for the general. He would be crown'd:
How that might change his nature, there's the question.
Brutus is fearful that Caesar's nature might change and that he could be overwhelmed by his power and importance and thus become a dictator, as he muses:
...that what he is, augmented,
Would run to these and these extremities:..
He concludes that Caesar should not be given the opportunity to achieve such power and, like a serpent in its egg, he should be destroyed.
Cassius expresses an intense hatred for Caesar and has a personal vendetta against him. Although he agrees that preventing the general's ascension would be for the greater good, his reasons for killing him display a greater personal slant. In his conversation with Cinna, he compares Caesar to a terrible storm caused by a man not better than they, who will bring untold misery and chaos. He rhetorically asks Cinna why Caesar should be allowed to tyrannise them.
His great hatred for the general is expressed in the following lines in the same scene in Act 2:
...what trash is Rome,
What rubbish and what offal, when it serves
For the base matter to illuminate
So vile a thing as Caesar!
In this regard, then, Caesar has to die and deserves nothing less. In the end, it is Brutus who comes across as the most rational of all the conspirators, when he says:
O, that we then could come by Caesar's spirit,
And not dismember Caesar! But, alas,
Caesar must bleed for it!
He suggests that they have no other choice but to kill the general for he believes that Caesar's spirit will not be swayed - he will be stubborn and refuse to listen to reason once he has gained supreme authority. In their plan to kill Caesar, he mentions that their act should not be seen as one performed by murderers, but purgers - they would cleanse Rome of an unnecessary evil.
That, in the end, is what all the conspirators believe. They are all acting on a belief in Caesar's possible malice and that is all there is to it. None of them provide any empirical proof that Caesar would, indeed, become a dictator once he is emperor.
It is ironic that after Antony's passionate speech, the conspirators are sought by the mob for being murderers. Their noble ideal has been shattered.
The conspirators believed that by killing Caesar they would be able to restore the Republic. They felt that Caesar was becoming dangerously close to a monarch. They had some warrant to think this, because Caesar took the title of dictator for life. In Roman society, the position of dictator was used only for a short period of time in emergencies. So, when Caesar became dictator for life, he was breaking with Republican customs, which were important to the Romans. In the end, they were mistaken. The death of Caesar did not establish the Republic, but ushered in an Empire under the leadership of Octavian.