1. Nero: Murderer; accused of setting fire to Rome.
Nero was accused of setting fire to Rome, because, as the emperor, he did little to stop the fire, described as playing his fiddle while Rome was burning around him.
2. Octavian: Defeated Antony and Cleopatra.
Antony had married Octavian's sister, but instead decided to sever the marriage to take up marriage with Cleopatra. Using this as justification, Octavian decided to dispatch of a potential rival for power, and declared war on Antony and Cleopatra. He was later victorious.
3. Caligula: insane.
Caligula is known for his insane actions, such as asking his army to collect shells from the seashore, or even to attack the sea.
4. Hadrian: Built a defensive wall in Britain.
Hadrian is well known for his Handrian Wall, which borders Roman territory Britain and kept out the barbarians.
5. Marcus Aurelius: scholar, philosopher, and soldier.
I don't know much about Aurelius, but since this is the one off, this would be the only answer choice. However, most Roman politicians were scholars and/or philosphers, and many an emperor had experience in the battlefield.
