Antony speaks these words in Act III, Scene 2 when he is giving a speech at Caesar's funeral. He is speaking ironically as he delivers this speech.
What he is saying here is that people only remember the bad things that other people have done. He says that we only remember the evil that the dead did and we forget the good. I do not know that this is true, but it is what he says.
He goes on to say that this is what they should do with Caesar. That's where the irony comes in -- he is really trying to get the people to remember the good about Caesar.
Marc Antony employs many statements with hidden meanings in his famous funeral oration of Act III, Scene 2, and "The evil that men do lives after them" is certainly one of them. This statement implies that history records the wrongs of people in more inflammatory words that are long remembered, while often their good deeds are either mitigated in the shadow of the more interesting evils or even forgotten.
Those who hated Caesar were eager to speak of his evil, Marc Antony hints in his statement; this act of suggestion is his subtle way of beginning to cast aspersions upon the conspirators. Soon afterward, Antony alludes to the accusations of Brutus and the others, but he again is subtle as he adds,
The noble Brutus
Hath told you Caesar was ambitious.
If it were so, it was a grievous fault,
And grievously hath Caesar answer [for]it. (3.2.79-80)
Thus, Brutus begins to sow the seeds of doubt into the minds of the plebeians who listen, so that when he reaches the end of his speech, the crowd will be eager for rebellion.