Starting with situational resemblances, Edmund and Antonio are both younger brothers who are angry and jealous that they are not in a position to inherit the titles and lands that rightly go to their respective brothers. Edmund cannot inherit because he is illegitimate, Antonio because he is younger. Both decide to take matters into their own hands and get rid of the people who stand in their way.
Edmund and Antonio are both characters will evil hearts. They are both plotters and planners who follow their own laws rather than adhering to a more universal moral code. When Sebastian ask Antonio about his conscience, Antonio replies,
"Ay, sir, where lies that?"
Likewise, Edmund shows his lack of conscience as he plans his brother's fall and betrays his father so that he can benefit and become earl:
"That which my father loses—no less than all.
The younger rises when the old doth fall."
Edmund shows some twinges of conscience as he repents of ordering the death of Lear and Cordelia and tries to stop it. Antonio, however, never feels remorse for what he has done. Both are evil characters who harm others to get ahead. Rather than trying to find success along a different path, both try to take what belongs to someone else.