You're maybe wrong about your assumption about conquering sophisticated people rather than great tracts of land, but you're certainly overlooking several important things.

Caesar didn't just conquer Gaul - and remember that the Gauls were Rome's historical bogyman. Caesar was born into a situation that put a spotlight on him. His uncle Marius was the only Roman before Caesar to hold seven consulships, six of them consecutively, to fight off the threat of the Teutones and Cimbri (a Gallic/Germanic confederation), and became known as the "third founder of Rome" for that accomplishment. Marius' party lost Rome's first civil war, and Sulla banned images of Marius from appearing in public. When Caesar's aunt, Marius' wife, died, he gave the funeral oration and unilaterally rolled out the images of Marius for the funeral. Caesar was 31. That was the year he was elected quaestor, the lowest rank on the cursus honorum. From this low position and sad occasion Caesar resurrected the memory of his uncle and made a direct connection between himself and Rome's third founder.

In 63, he was elected to both the office of praetor - one step away from the consulship - and the pontifex maximus - Rome's highest religious position. He borrowed heavily to bribe voters, and had to sneak out of Rome to avoid his creditors to take up the governorship of Spain, but he won.

His civil war was not fought only (well, mostly, but let's not get complicated about it) in Italy like Sulla's was. Caesar's war was fought in Italy, Greece, Egypt, Africa, and Spain. When Caesar celebrated his triumph, it was the most lavish triumph ever celebrated up to that point. It is known as the Quadruple Triumph, celebrating victories in Gaul, Egypt, Africa, and Pontus (that's the "veni vidi vici" campaign). You can read about the extravagance of the Quadruple Triumph in Appian here.

He DID NOT change the constitution of Rome from a republic to a dictatorship. Dictator in the Roman system was a constitutional position; he simply applied it in new ways, and he wasn't the first to do so. That was Sulla. Caesar WAS the first Roman to put his own face on a coin. His affair with Cleopatra was a famous scandal, and had political importance. Cleopatra was the last of the Hellenistic monarchs, and Egypt was incredibly rich. When she gave birth to Caesar's only male child (and living child, since Julia died ages ago), there was a real risk of an international crisis, depending on whether Caesar made the kid his heir.

I'll skip the details of the events following the civil war and the assassination. Here's the kicker: he had really good press after he died.

Octavian, on hearing he was Caesar's heir, came to Rome with no levers of power. He used his connection to Caesar to get leverage against Marc Antony first, and then against the assassins, and then against Marc Antony again. He stressed pietas, one of the most important Roman virtues, as the driving force behind his project for killing the assassins. During the course of his reign, he promoted his connection with Caesar, and also constructed parallels between himself and Aeneas, who also was famous for his pietas, in sponsoring work like the Aeneid. If you want some scholarship on how Octavian (AKA Augustus) did this, you might start with The Power of Images in the Age of Augustus by Zanker.

I'm not able to say whether Caesar would not be iconic without Augustus' efforts at popularizing him, but I am certain that Augustus set out on purpose to make Caesar an icon, as Augustus' connection to Caesar was central to his own political power. And he was quite successful at it, hence Caesar became a title rather than a name all the way into the 20th century, as you note.

Having a Shakespeare play helps!

The most direct answer is "he adopted Octavian" who ultimately came out on top of the tumult of the Late Republic and thus gave the family name "Caesar" to his successors. Augustus was himself active in promoting Julius Caesar's posthumous reputation, although of course in the Roman world Augustus was always the more important.

Beyond that, the last century or so of the Republic was generally considered a very exciting time with colorful figures, a Caesar forms a sort of capstone to it (for example, of the twenty Roman biographies in Plutarch, thirteen are of the period beginning with the Gracchi). Giving an exact reason why particular periods are so prominent in popular imagination is always going to be difficult to explain rather than justify--to give a comparison, why is the Tudor era so prominent in British history? You can give justifications: Shakespeare, the drama of the English Reformation, beginning of the overseas empire, the Spanish Armada, etc. But you could also give justifications for why, so the period of the Civil Wars, Commonwealth, and Restoration could be just as dramatic--you have Milton, the wars themselves, the cloak and daggers intrigue of the Commonwealth, the enigmatic figure of Cromwell, etc. Some periods--the Three Kingdoms in China, the the early Medici in Florence, the piracy of the late 1600s, etc--tend to "stick".

And of course fame tends to snowball. Because Caesar was famous, individual stories about him could be told and retold in various different ways for various purposes. His capture by the pirates is the ultimate example of keeping it cool, calm, collected, his assassination is either the acme of treachery or a stirring example of taking arms against tyranny, he can be woven into the national myth making of France and Britain. And the more these references are used, the more they can be used, and the more of an icon Caesar would become.

I strongly recommend Maria Wyke's Caesar: A Life in Western Culture which takes various episodes in his life and shows how they have figured into culture in later periods, from Caesar as chivalric ideal to Caesar and Cleopatra decadence and luxury giving name to a Las Vegas casino.

There are two excellent points that are brought up and that I will concede may be more important than my own. 

First up, Julius Caesar was a winner. That counted for a lot. We can argue about what kinds of victories were preferred but you had to win, and he did. u/LegalAction explains that and much more here.

Secondly, Julius Caesar was succeeded by men who glorified him rather than tore him down. This is always vital to a historical reputation. It is not so difficult to imagine a world where the alleged "republicans" of the civil war following Caesars death (conspirators or liberators may be preferred) won that war and built a story about how evil Caesar was to match the good PR he got from Augustus.  u/Tiako explains it well here.

However, there is another thing that can cause a man like Caesar to stand out. He was genuinely different from his contemporaries and his successors in his approach to power. By the time he entered public life, the republic had been rocked by civil war before and the time tested methods of confiscating property and killing enemies (and potential enemies) had been trotted out. Sulla had led the first proscriptions, a horrific event where names of public enemies were hung in the forum and anyone could collect a bounty by murdering them and bringing in the head as proof. This practice would be revived by Caesars successors, but he never approached anything like it himself.

This was not to say that Caesar was a teddy bear. He was ruthless about employing violence and cruelty when it was pragmatic. Perhaps the most famous is during the siege of Alesia, where he left the residents of the settlement who were ejected starve to death in the zone between the walls and his own fortifications. He was not a stranger to violence. However, he did not like to employ it against fellow citizens.

This started with his speech during the Second Catilinarian conspiracy, an affair that happened during Cicero's consulship. A group of people were plotting the violent overthrow of the republic and the Senate discussed whether or not they should be executed. Caesar was the first to propose that they should not be. His rather nonsensical solution was to imprison them in allied Italian towns, since Rome had no prisons. The Senate decided to execute them, probably because the solution was wholly impractical, but he had proposed clemency.

This tendency towards mercy was something that others did not always appreciate. Famously, Cato the Younger killed himself rather than be pardoned by Caesar at the climax of the African campaign. Caesar was killed by men he had pardoned for waging war against him. He pardoned men who left only to take up arms against him immediately, to the point where his own men found it utterly ridiculous. While besieged in Alexandria, he pardoned Ptolemy XIII, who supported the besiegers the second he left the Roman lines. 

It is hard to know the motives of a man two thousand years gone, but I tend to agree with the speculative notion that what Caesar wanted was to change the way Rome fought civil wars. Caesar didn't imagine that his victory would end further civil wars, and he probably thought civil war was a kind of life insurance for himself. Surely everyone would think his rule was preferable to another round of civil war. He was wrong, obviously, but I think the hope was that Rome would stop culling all of its best men every generation. If the norm could go from execution to pardons, the senatorial class would stop removing it's most capable members. I'll freely admit that's conjecture, because again, it's hard to know what Caesar intended or what kind of system he would have set up had he lived.

If we take Caesar as one Roman warlord among many, which I don't think is a big stretch, he would be typical in many ways. Was he successful militarily and politically? Yes, but others were as well. It would be an argument of kind rather than degree. But, the fact that he was stabbed to death by men he had pardoned and who had done quite well under his rule was unique. It was a justification for future acts of murder by many roman emperors, but killing rivals and potential rivals is simply a "best practice" of dictatorship. Even if Caesar's murderers had won the war his death caused, I still think he would be famous because he was so different from other Roman warlords in regards to his use of clemency over ruthlessness.

This is drawn from Adrian Goldsworthy's work, whom I have been binging. Primarily Caesar, Life of a Colossus and How Rome Fell: Death of a Superpower.