I remember from Latin class that several emperors were killed by the Praetorian Guard for just being too bad to let live. I'm curious if these executions were carried out secretly or publically, if the assassins were acting through the will of another entity, and what sources we have available to study these assassinations.
It depends on who the assassins were, and why they did it. In many cases, the Praetorian Guard or the army were responsible. The practice was often rewarded.
Only two punishments spring to mind. The first is Cassius Chaerea, a Praetorian who killed Caligula. He did so as part of a wider conspiracy involving several senators and members of the Guard. According to Suetonius, he was sentenced to death under the rule of Claudius.
Domitian's murder was probably also the result of a large conspiracy. Petronius Secundus was the prefect of the Praetorians at the time. After Nerva ascended to the throne, there was a bit of a revolt, and the Praetorians killed Secundus. I'm not sure if he was responsible or if the charge of murder was political.
In the 3rd century, assassination was pretty much the mechanism of succession. Commodus was killed by a concubine and wrestling coach. I'm not what happened to them. At this point, the politics get messy for nine decades and primary sources get spotty. In several cases, all we know is "killed by the Praetorian Guard" or "killed by the army." Our best source for this period is Herodian.
If we believe Vopiscus, Numerian was most likely poisoned by his father in law Aper. Aper was publicly killed by Diocletian. Again, Aper's death was as much a matter of political succession as it was punishment.