It depends on the time and place, really. In the early days of the empire, Romans balked at worshiping a living emperor but people living in the eastern half of the empire worshiped him as a matter of course, they had been worshiping their rulers for centuries.

In the city of Rome there were two different ideas about worshiping the emperor that worked in tandem. First, the Romans believed that every man had a genius, the closest thing we have to it is a soul but for Romans it was kinda also like a guardian angel, a divine double. It was part of you but also existed on its own and guided you. The genius of the paterfamilias was an item of worship in the household cult. By extension, the genius of the emperor the pater patria was worshiped in the civic cult.

The other tradition was apotheosis after death. Roans really only worshiped their emperors (as opposed to their genius) after death. The most entertaining example is on the Arch of Titus, which is located at the entrance to the Roman Forum. On the top of the underside of the arch is a depiction of Vespasian (Titus's father) being carried to heaven on the back of an eagle. Vespasian's last words, following in a long career of wry wit, were reportedly "Alas, I can feel myself becoming a god."

/u/mp96 has a great write up on this here