How was a leader chosen in ancient rome? How does a leader arise in ancient rome?

Answer 1

Two "consuls" were elected annually by the Senate and the People of Rome (SPQR) in the days of the Republic
(~509 to ~ 27 BC). They were almost always from the senatorial /aristocratic class.
When Augustus Caesar established the Empire with an emperor in charge (after 27 BC), it was usually the previous emperor who decided his successor. Consuls then became just figureheads.
In ancient times before 27 BC, the consuls were the generals of Rome's armies, so in time of war it would usually be the most renowned soldier/senators chosen as co-consuls. What was strange is that they would each command the army on alternate days (to prevent abuse of power and tyranny which Romans feared). So one day Consul "A" would be in charge, and the next day Consul "B" would be giving the orders. At least that's the way it was in ancient times such as the time of the first two Punic wars with Carthage in the 200s BC.
[ Wiki is fairly OK on this topic]

Answer 2

Spreedog133's answer regarding the succesion of rule during the principate (empire) is generally incorrect. Although succesion by choice of the previous emperor was the ideal model, it was rarely the case. Most emperor's were installed by the military forces they commanded, and many of them did not last particularly long on the throne.