Antony sends Lepidus to Caesar’s house to get the will so they can determine who should get what.
At this point in the play, a lot has happened. After Caesar’s death, Mark Antony made sure to establish himself in the hearts and minds of the people as the new leader of Rome. There was a problem though. Caesar named Octavius his heir. Also, militarily, Antony did not have the full force of the army.
This is how the triumvirate was formed. It was a power sharing arrangement that had to do with expedience during the days after Caesar’s death when Rome was trying to figure out who would actually be in charge.
In this scene, we see that Antony is ruthless and harsh, Lepidus is considered a dolt, and Antony and Octavius are the ones vying for power. Antony is able to get Lepidus to agree to just about anything. He knows that he is the weak point in the triumvirate. He was only allowed in because he had an army.
The purpose of this meeting is to discuss the proscriptions. This was a nasty business, in which they will take property from or kill enemies. The enemies might be conspirators, or just people they want killed. Lepidus agrees to kill his own brother! (Of course, Antony agrees to kill his nephew.)
Antony turns his attention next to the will.
But, Lepidus, go you to Caesar's house;
Fetch the will hither, and we shall determine
How to cut off some charge in legacies. (Act 4, Scene 1)
This is important for two reasons. He wants to see where he can limit the amount of money certain people get. First, remember that he used this will to show that he was important to Caesar and to remind the people that he would carry out Caesar’s wishes. Now he is breaking his promise. Well, war is expensive. Second, he is basically making an errand boy of Lepidus. Lepidus’s reaction is noted.
What, shall I find you here?
Or here, or at the Capitol. (Act 4, Scene 1)
They are pretty much reminding him that he is a third wheel. Come and find us, we’ll be around. Just look for us, okay? You are not important enough for us to send a messenger telling you where we will be, and you are not important enough to be here so we are just making you our errand runner. Ouch.
The battle for power between these three will continue for some time, until it ends rather spectacularly with the war between Octavius and Antony in Antony and Cleopatra. Lepidus does eventually get shunned aside, accused of treason. Was Lepidus guilty of betraying Octavius? Probably. Shakespeare makes him seem like a somewhat innocent victim, starting right here.