The plebeians certainly do not think of Caesar as a tyrant at the beginning of the play. Most of them regard him as a hero and a potential benefactor. The conspirators do not regard Caesar as a tyrant but as a potential tyrant. Brutus expresses that view in his soliloquy in Act 2, Scene 1:
And since the quarrel
Will bear no color for the thing he is,
Fashion it thus: that what he is, augmented,
Would run to these and these extremities;
And therefore think him as a serpent's egg,
Which, hatched, would as his kind grow mischievous,
And kill him in the shell.
Brutus, characteristically, is already thinking about the speech he knows he is going to have to make after the assassination is accomplished. He is always concerned about what people think of him. His speech sways the plebeians in Act 3, Scene 2; and this is where the words "tyrant" and "tyranny" come up.
FIRST CITIZEN: This Caesar was a tyrant.
THIRD CITIZEN: Nay, that's certain.
We are blest that Rome is rid of him.
They obviously have been strongly influenced by Brutus's speech. Antony will have a hard time turning them around, and he knows it. He may have heard what one of them said:
FOURTH CITIZEN: 'Twere best he speak no harm of Brutus here.
As far as the opinion of the conspirators is concerned, one of the minor characters makes an appropriate statement right after Caesar's assassination:
CINNA: Liberty! Freedom! Tyranny is dead!
It seems pretty obvious that both the real Julius Caesar and Shakespeare's Julius Caesar would have become tyrants in the future.