There was no concept of "Emperor" or "Augustus" for that matter. The man who would become Augustus was the 19 year old grand nephew of Caesar. Caesar was never crowned a king, despite what everyone thought was his intention. He was a senator who amassed unprecedented constitutional power through war and popular politics. He was "dictator for life" but not Emperor, the concept of an actual Emperor never existed in absolute explicit terms, at-least not until the third century AD or later. It would take Octavian over a decade to consolidate power and achieve his dominance, a dominance that was mostly implicit. There was no declaration of absolute power, he beat his enemies and held power through patronage, wealth and a rotating series of political appointments. Octavian contextualized himself as the savior of the republic, not it's destroyer. "Emperor" is a historical simplification of a more complex political reality.
The issue is with some of the fundamental assumptions of your question -- Caesar had no monarchical rulership which transferred automatically to his heir. He possessed the dictatorship upon his death, but this is not inherited. Caesar did have an immense amount of wealth, of which the mass was supposed to be inherited by Octavian (Augustus), and Octavian made good use of the political leverage of being Caesar's (adopted) son. There was over a decade of infighting, real fighting, alliances made and broken, and other such tension involved before Octavian assumed the role that we call emperor, and he was careful to distinguish it from the unpopular aspects of the dictatorship held by Caesar. Goldsworthy's biography of Augustus is a good place to start on how complicated this process was. In short, though, yes, it is a fair understanding that the conspirators overestimated their ability to control the situation in the wake of the murder. Probably the prime difficulty in their plans was Marcus Antonius (Shakespeare's Antony) and his opportunism.
The wording of your question seems to imply that the assassins would have realised that the murder of Caeser would lead to the ruler by emperors. In fact this was not the case. No one knew that Octavian/Augustus would be adopted as his heir in his will. Octavian was only 17 at the time of Caeser's death and had not even begun his political career yet; he was a political nobody at the time of Caeser's death.
The republic worked through a system of power sharing. There was a system of magistrates, from the junior elected military tribunes all the way up to the consuls, of which there were two elected each year. The system allowed Roman politicians to exercise their extreme ambitions, but through its checks and balances meant that glory would always be a shared thing - no man would ever be forever in the spotlight. But, after his victories in the civil wars, Caeser brought so much of the normally shared powers all under his own person that he cut off other aristocrats and politicians from pursuing their careers in the way that roman tradition had accustomed them to.
The republican system was already breaking down before Caeser's dictatorship, with violence becoming a cornerstone of politics in his generation, and with men like Pompey flouting tradition at every turn. But Caeser was the first to take it as far as he did.
The assassins seemed to believe that after Caeser's death the republic would return to normal. By assassinating Caeser, all his partisans and allies would also become politically disenfranchised, so removing Caeser would also remove all the men brought to power through him. Naturally, the assassins would all benefit politically from this, as it would mean removing so many of their rivals.
The Roman heritage brought with it a deep-seated hate and distrust of monarchy, and the assassins thought they defending the republic from a would-be monarch. They misjudged the situation - the Roman public did not want Caeser's death. In the aftermath of the murder, they did not immediate purge Rome of his partisans - most importantly, Mark Antony. They failed to take full advantage of the situation afforded them after Caeser's murder, and this allowed their enemies - namely, Mark Antony and Octavian - to consolidate Caeser's political, material and military legacy and, eventually, take over the Roman state.
Using for my source Caeser, Adrian Goldsworthy.
It's a long and complicated answer, and I'm currently on a phone so I apologize for typos.
When Caesar dominated the Republic, he was doing something mostly unprecedented. Since 509 b.c.e, Rome had been a republic ruled by popular election with the notable exception of the "dictator" position. During a time of existential emergency, the two popularly elected consuls (think US President) could appoint a man to take complete control of the government in order to run the state in a hyper efficient manner. The most famous example of this is Cincinnatus (wiki if you would like more background).
For almost 400 years, succession of power was passed via popular election with a few hiccups. For more background on the hiccups, read Sulla and Marius.
Caesar's rise to power, contrary to popular belief, was not likely due to avarice for power. After a wildly successful campaign in Gaul (France), the Senate ordered Caesar to disband his legions. However, he knew that if he were to do so, the senators would arrest him and try him for crimes committed during is campaign in Gaul (namely he stayed abroad too long, not wanted to cede his progress made). Therefore he made the decisions to leave his army intact and march on Rome.
After defeating various rivals, Caesar found himself in complete control of the Senate. However, should he ever cede control for one minute, he knew a large contingent of senators would try him and have him executed. He had essentially gone too far and had no choice but to keep going. So, he declared himself "dictator" for life so that he could bypass pesky things such as term limits.
Eventually the Senate killed him. Et tu, Brute. Now we can finally get to the heart of your question. Upon Caesar's death in 44 b.c.e., Rome was still a Republic in every sense of the word. After Caesar nothing changed legally. The people still held elections and the public offices remained the same. But, there was a massive void of power which three men seized upon (Octavius, Marc Anthony, and Lepidus) and essentially took turns as the "official" leader of Rome. This triumvirate disintegrated and civil war broke out which lasted until 31 b.c.e. So, to answer your question there was a 13 year civil war period between Ceasar and Augustus which the Senate likely didn't anticipate. But there certainly wasn't a direct passage of power like the European monarchs have.
It was Octavius, given the epithet "Augustus" in 27 b.c.e., who LEGALLY went about the long, slow process of transferring all the power previously held by popularly elected offices to a new office, commonly known as the emperor. After he had legally created this office, it could then be passed down via hereditary.
Probably a far longer answer than you were looking for, but there you have it.
I'd like to add something to /u/QVCatullus' and /u/i_like_jam's perfectly correct answers. Augustus' becoming emperor didn't happen overnight, nor was it really the result of his adoption by Caesar at all. Octavian became the sole leader of the Roman political sphere only through about a decade of extremely bloody struggle, both political and military.
In addition to what they said, I'd like to note that first of all nobody expected Octavian to be Caesar's heir, and second of all the conspirators were hardly as united in purpose as you seem to think. Caesar's original will almost certainly stipulated that Mark Antony was to become his heir, and he had it secretly changed when Antony proved himself totally incompetent during his governorship of Italy. Octavian was a nobody, only distantly related to Caesar by an undistinguished line of an undistinguished family, with no political connections whatsoever and, apart from a brief period when he joined Caesar during the Spanish campaign, no experience outside of his home. When the conspirators murdered Caesar it was with the expectation that Antony would become the leader of the Caesarians. It's important to note that that's what being Caesar's heir meant. Octavian inherited no political functions whatsoever--in fact, for the early years of his career, immediately following Caesar's death, he held no official position at all and was forced to build power through his social interactions with various nobiles and Caesarians, as well as (illegally, since he held no magistracy) building an army of Caesarians largely stolen from Antony's ranks. What Octavian inherited was Caesar's official blessing to preside over the ranks of the Caesarian faction, which gave him access (at least in theory) to Caesar's vast network of political and social connections. However, this can easily be overstated, since political alliances were far from hereditary and even patron-client relationships were by no means necessarily passed down from generation to generation. What Octavian inherited (or rather, did not inherit, since Antony disputed the authenticity of the will and refused to comply with it) was Caesar's wealth, which included the ability to pay the Caesarian legions. Caesar's wealth was enormous and was one of, if not the, most important political tools in his arsenal by the end of his career.
The fact that Antony was expected to inherit all of Caesar's stuff only raises further questions, though. If Octavian was a nobody then the conspirators surely would've felt they had nothing to fear from him, had they known about the actual contents of Caesar's will. Antony, on the other hand...But Mark Antony was hardly the man he portrayed himself as being later when he was a Triumvir. During the Triumvirate Antony liked to style himself as having been Caesar's most trusted lieutenant and his right-hand man, a pretension that Cicero and other sneered at. Antony was nothing of the sort. While he had been a competent military commander (though not brilliant--unlike with Labienus, Caesar never found enough trust in Antony's ability to give him the freedom that he gave Labienus, who essentially fought entire campaigns in Gaul by himself, without Caesar's input at all. I have little doubt that Labienus' defection to Pompey played no small part in Caesar's unwillingness to rely too heavily on individuals and to give them too much power, but even in Gaul, before Labienus' defection, Antony was put on a fairly short leash) he was a complete fool of an administrator and a pretty piss-poor politician--as his total failure to administer Italy during Caesar's civil wars and his total bungling of the political situation after Caesar's death attest to. Antony's failure to govern Italy seems to be what really sounded the death knell for his inheritance of Caesar's power--after all, he had permanently alienated part of Caesar's most devoted supporters, forcing Caesar to return prematurely from his campaign to fix the problem. Antony's public disgrace, being stripped of all titles and honors, was combined with Caesar's reluctance to grant him any political positions for the rest of his life. Still, in the last couple years of his life Caesar was beginning to warm back up to Antony a bit, giving him his honors back and allowing him to hold office, so for the conspirators it was hardly as though he was not going to be a problem to handle.
Which leads us to my point about the conspirators not being as united in purpose as we would like to believe. It's quite clear from reading the accounts of the conspiracy, both those of later writers and of contemporaries like Cicero, that the conspirators had a variety of political motives and came from a whole bunch of political camps. The fact that Brutus, a (reluctant) Caesarian with very close personal ties to Caesar (and a man who owed Caesar his life), was a part of the conspiracy shows this quite well. And the conspiracy was hardly restricted to the 21 men who actually pulled out daggers and plunged them into Caesar's chest. It encompassed political figures throughout the Roman state, to such a wide degree that Caesarians (most notably Antony) started accusing people left and right of being in on the conspiracy. And Antony's role in this should not be overlooked. While he was blowing lots of hot air accusing people like Cicero (who defended himself brilliantly) of being conspirators it must be remembered that this is something that he was duty-bound as the de facto leader of the Caesarians (and as the man who refused to hand over leadership to Octavian) to do. There may have been another motive as well. I'm inclined to agree with Badian and others who suggest that Antony might have been a part of the conspiracy. Whether we buy that or not, he certainly knew about it--every important politician in Rome knew that something was up. Antony claimed that he had only heard about the conspiracy a day before it happened and just so happened to have gotten there a moment too late, but this seems unlikely. First of all, Antony was closely tied to several members of the conspiracy, which included several notable Caesarians. And Antony, as the most influential of the Caesarians (and the man to whom, after Caesar, the army was most loyal) would've undoubtedly been courted, as he in fact had been during Trebonius' conspiracy against Caesar earlier that year. That Antony rejected the offer and exposed Trebonius' conspiracy does not necessarily mean that he did the same thing with the second round, however, since Antony's actions against Trebonius were in part what reconciled him with Caesar, less than a year before his murder. So Antony certainly knew of the plot, and he was the guy, as Caesar's apparent heir, who would gain the most if Caesar were removed, just as Antony's career was turning around. If Antony was a part of the conspiracy, or even if he knew about it and was willing to turn a blind eye, the conspirators may well have felt safe in their actions.
The point of bringing up Antony, whose part in these affairs is still widely debated, is to show just how enormous a conspiracy this was, and that not only were all kinds of people involved, but there were lots of people who knew about the plot and did nothing to alert anyone. Cicero, for example, was accused by Antony of being a part of the conspiracy, and although he defended himself quite well we know for a fact that he knew that there was a plot of some kind against Caesar, and pretended in public that he knew nothing about it. The conspiracy against Caesar is an awful lot more complicated than simply saying that a bunch of senators were pissed off that Caesar had so much power. Sure, some of the conspirators fall into that group, but there were a lot more motives involved