
John G. answered 02/04/20
Historian Experienced in European History
Your question is very vague, and depends on not only the time period, but what constitutes "people" in a given circumstance. For example, are we referring to the height of the Republic? The civil wars of the first century BCE? The empire? Are they running for election to a magistracy? If so, which ones?
We'll start with a properly-functioning Republic. Politicians running for office needed to win the votes of a wide range of people - while the aristocracy was the most powerful voting block, they could not ignore the lower classes, either. Among the upper classes, the most important thing was personal relationships, including marriage ties, political alliances, and friendships; tapping into this network well (and exploiting the patron/client relationships that formed the core of what we tend to refer to as "political parties" in Rome) helped you get elected fairly easily. Gaining the support of the lower classes, on the other hand, tended to involve a lot of generosity, including but not limited to the sponsoring of festivals and performances. Additionally, successful military endeavors and oratorical skill helped you gain support at all levels of society.
During the latter years of the Republic, amidst the periods of the civil wars, you had to appeal to three groups simultaneously: the aristocracy in the Senate, the lower classes (who actually passed legislation, not the Senate), and the premier generals of the day (Sulla and Marius, Caesar and Pompey, Octavian and Antony). The most successful politicians in this era had the patronage of a general and were well-liked by the lower classes, but were not so closely linked to either that they were not drawn up into the violence of the civil wars, simply remaining members of the Senate. Support the wrong general, and you ended up dead in the proscriptions, killed by the supporters of the victor. Anger or terrify the Senate too much, and you ended up assassinated. Lose the support of the people, and you found yourself caught up in mob violence. In short, you had to keep everybody happy.
Not surprisingly, the Empire saw a different means of politics. Although the facade of the Republic was maintained, appealing not to the populace or to the Senate, but to the Emperor, was most important, as the Emperor was the one who dictated who was allowed to run for office, who was allowed to host festivals, and who was allowed to propose bills to the Senate - and what those bills were.
All in all, as we've seen, the politics of Rome varied greatly throughout its long time. I hope this brief rundown helps!