Final answer:
Aemilianus was the emperor who declared himself after defeating the Goths and assassinated Trebonianus Gallus, but was killed by his troops in favor of Valerian after only 3 months, during the Crisis of the Third Century.
Step-by-step explanation:
The Roman Emperor described in the student's question is likely Aemilianus (Marcus Aemilius Aemilianus). He was proclaimed emperor by his troops in 253 CE after a victory against the Goths.
Aemilianus then marched on Rome and ousted the sitting emperor, Trebonianus Gallus. However, Aemilianus' rule was short-lived; after only three months, he was assassinated by his own troops who switched their loyalty to Valerian when he arrived with a larger army.
This period of Roman history illustrates the turbulence of the third century, marked by rapid changes in leadership and external threats, known as the Crisis of the Third Century.