Did run pure blood; and many lusty Romans Came smiling, and did bathe their hands in it.
I think that the Romans washing their hands in the blood of Caesar symbolizes the guiltiness of the Roman people. The Roman people do not want Caesar to have the crown because they fear he will turn his power against Rome despite him not having a past full of abuse of power. In response to this fear they want to kill him before he potentially abuses his power and destroys the fate of the city. In reality though, the city is in greater danger if they kill Caesar and their blood is in their hands. Except its not just Caesars blood in their hands it's Romes blood too.