In the film, Marcus Aurelius says that Maximus is the “son that I should have had”. He knows that his biological son, “Commodus is not a moral man. Marcus says, "Commodus cannot rule. He must not rule.” They know that Commodus has displayed some of the most deplorable traits possible for a human being.

Commodus is a brilliantly conceived and developed character and villain in the film. From his actions, it is easy to see that he is definitely insane and deplorable. But, he clearly explains his motives and actions. He is psychotic; he is devious. He smothers and murders his father with a pillow before power can be transitioned to Maximus. Thus, the rules of succession automatically transfer the throne to Commodus. He is to be the next Emperor of Rome!

Before Marcus is killed, he summons Commodus to his chambers after his talk with Maximus about his plans for succession. Marcus knows his son will be devastated by the decision. Upon hearing the news, Commodus unsurprisingly breaks down and tells his father “You wrote to me once, listing the four chief virtues: Wisdom, justice, fortitude and temperance.

Commodus says to his father that he had none of his father's four virtues. He has other virtues. Ambition could be a virtue if it caused one to excel. "Resourcefulness, courage, perhaps not on the battlefield, but … in many other forms. Devotion to the family could be a virtue. But none of my [Commodus's] virtues are on the his father's list.” So, his father must die.

More Info: en.wikipedia.org