Fyodor Pavlovich was killed by the servant Smerdyakov, not by Mitya, who is convicted in his place. The book does not show the murder: it recounts it afterward, in the confession Smerdyakov makes to Ivan, on the eve of the trial.
Smerdyakov says he feigned an epileptic fit, waited for Mitya to appear, and, when he saw Grigory lying in the garden and understood that Mitya had been there, tapped the agreed signals on the windowpane. The old man opened, and he struck him three times on the top of the head with an iron paperweight. He took the money from behind the icons, hid it in a hollow in an apple tree, and went back to moaning in bed. He calls Ivan the principal murderer and himself the accomplice, because Ivan left for Moscow when he could have stayed: he read the departure as consent.
That same night, after Ivan's visit, Smerdyakov hangs himself, leaving a note that says he kills himself of his own will and pleasure, so as not to blame anyone. In court, Ivan presents the money and declares himself the murderer; the doctor declares that he is feverish and hears him speak of the devil, and no one believes him.
Mitya, for his part, is not innocent of everything: he wanted to kill his father, threatened him in front of witnesses, and was in the garden with a pestle. But the novel does not narrate what happens between the instant he raises the pestle and the moment Grigory falls. That ellipsis is what leaves the jury, and the reader, to decide with the facts they have.