How is the trait of fate shown to Brutus and relates to him being the tragic hero?
The most notable point at which fate is revealed to Brutus in Shakespeare's Julius Caesar is the point at which Caesar's Ghost appears to Brutus in Act 4.3.
This can only be a bad omen. The Ghost appears and, while the stage directions announce it as the Ghost of Caesar, the Ghost itself tells Brutus that it is "Thy evil spirit." Then it tells Brutus that it will see him at Philippi, the eventual battlefield.
The implication is, of course, that the Ghost, and Caesar, will get revenge at Philippi. Brutus's fate is already sealed.
Of course, the place of the final battle is left up to Brutus, who, as the tragic figure, makes still another bad decision. Cassius correctly suggests that their armies remain where they are and defend their position, as opposed to moving and attacking. A defensive position is almost always the stronger position during a battle. But instead of listening to Cassius and giving their armies a better chance of winning, Brutus orders the armies to move. Once again, Brutus's decisions lead toward the downfall of the conspirators.