Just for the record, "Kaiser" means "Emperor", not "King". "King" in German is "Konig" and comes from a non-Roman root.

Originally Caesar was a family name. The Julii Caesares were a patrician Roman family that had long ancestry, but not much money or power for most of the Republic, until the great general Marius married into their family a few decades before Julius Caesar rose to prominence. Julius Caesar adopted Octavian (Augustus), so he got Caesar in his name, too, since that was how Roman nomenclature worked. For the first emperors, the Julio-Claudian dynasty roughly related to Augustus through birth, adoption, or marriage, Caesar was still essentially a name. By the mid-first century CE the emperors weren't descended from Augustus anymore, but by assuming the name Caesar they could establish a strong link to the earlier emperors and to Julius Caesar himself, and over time this became a standard part of the emperor's name. Christopher Mackay's A Military and Political History of Rome is a pretty good source on the rise of Caesar (as well as many events before and after).

All of the emperors were also Augustus, a title belonging to the person who takes the auguries (essential for choosing things like when and where to send an army). In the Greek part of the Roman empire, Augustus was often rendered as "Sebastaios."

Later emperors were constantly associating themselves with Augustus, though. Compare Augustus's mausoleum with Hadrian's, for example (the circular part). They didn't need to do this through names, necessarily, though; architectural projects worked great. Augustus finished off the Forum Iulium in honor of Julius Caesar, and many many emperors afterward added fora to monumentalize their legacies within the city, for example Trajan's massive forum project complete with Greek and Latin libraries, a huge column depicting his victory over the Dacians, and maybe some market space as well. Julian Bennett has a good bio called "Trajan: Optimus Princeps" if you're interested in this dynamic.