Upon viewing the play for the first time in 1599, the audience, well-versed in biblical tales, would have got the allusion of handwashing as a reference to Pilot and Jesus. Where Pilot uses water and "washes his hands" of Jesus and takes no responsibility for his actions and Jesus’ fate, Brutus, in contrast, takes full responsibility for the murder, ("..let us bathe our hands in Caesar's blood/Up to the elbows...!") and Casca urges the other conspirators to do the same. In III.1, Antony acknowledges those responsible ("Let each man render me his bloody hand") and implies he is now in league with them since in shaking each hand he has bloodied his own. ("Therefore I took your hands....friends I am with you all.") However, Antony reveals his true intent as Caesar's avenger several lines later when he is alone (Woe to the hand that shed this costly blood!) and describes the bloodshed to come.
By "bathing" in Caesar's blood after the event, they are taking equal resposibility for cleansing Rome of its overly ambitious leader. However, it does not play out the way Brutus and the others expect it to. Of course, this is partially because only Brutus has acted for the good of Rome alone and not for any other reason such as vanity, jealousy, or ambition. As a result, the conspirators are taken down by Octavius and Mark Antony--the blood covering the conspirators is a bad omen forcasting their own downfall and destruction.