How did the conspirators execute their …

First, they plan to have Decius go to Caesar's house to make sure he goes to the meeting. It's a good thing they decided to do this because Calpurnia almost has Caesar convinced to stay home, but Decius appeals to his vanity by telling him that people will say he is ruled by a woman's fears and by telling him that the senate plans to ask him to be king (see Act II, scene 2).

Second, once Caesar arrives at the meeting place, the conspirators surround him so that no one can warn him of what is to come or protect him when it happens. They make sure Antony is not around when they do the deed (Act III, scene 1).

Then they have Metellus ask for a pardon for his brother, who has been banished from Rome. That is their pretense for showing Caesar's unreasonableness and tyranny. When some pretend to bow to him, Casca raises his dagger and says, "Speak, hands, for me!" And they all proceed to stab him to death, proclaiming that they have done a good deed for Rome and its citizens.