The Roman dislike of kings (rex) goes back to the pre-Republic days when Rome was ruled by them and the last (according to Livy) was so terrible that he started a revolt which turfed him out and resulted in the formation of a Republic that was designed to prevent sole power being wielded by one man through institutions like the dual consuls and the senate and the populus Romani via elections. So the short form is being accused of aspiring to kingship is the political equivalent of being accused of being a communist in America in the 1950s. It's not good.

To understand Augustus' monopolization of sole power you have to understand the setting of the Late Republic. The system had been in a prolonged period of slow collapse. Factions had formed that were intractable, political violence was becoming more and more common, and there had been several bloody civil wars. Hundreds if not thousands had been killed in politically sanctioned murders. The last few decades of the Republic was either street fights between political gangs or outright open civil wars. Shit was on fire, yo.

So you have Gaius Iulius Caesar Octavianus (aka Augustus), the adopted son of Julius Caesar and inheritor of the powerful Caesarian political faction. Most of the powerful men who had the stature to challenge him for power had been either murdered after the formation of the Second Triumvirate or on the battlefield at Phillipi. The only person who was a potential threat to him was his supposed ally Marcus Antonius (aka Mark Antony). He was Caesar's lieutenant and he was controlling the Roman east in their agreed upon division of the Roman empire.

Despite their alliance in destroying the assassins of Julius Caesar and the Republicans, a new cold war sprang up between them. Octavianus actively propagandized against Antonius and built up public support against him with accusations of being dominated by Cleopatra and going 'native'. Eventually this cold war turned hot and Antonius was killed and the only man standing was Octavianus.

For the Roman public, they'd been in and out of public disorder for decades, their supply ships had constantly been interrupted by the sons of Pompeius (fighting for the Republicans and later as their own faction) leading to high food prices and some starvation. Finally Octavianus is bringing peace. Stability. Order. The senate won't argue, it's been decimated by the wars and refilled by men chosen by Octavianus - they're not going to raise a stink against their patron. Or they've survived by keeping their heads down and they're not about to change that when Augustus is the only authority in town.

What Augustus does next was quick clever. He saw from the example of his adopted father that his actions had garnered considerable unpopularity. His perpetual dictatorship got him assassinated. So he engineers through his agents and using the immense political power he commands gets several laws passed.

Firstly he gains the power of the tribunes of the plebs making him the defender of the people. It's great PR but it also removes the thorn of ambitious tribunes and their veto. Now only he wields the veto which is the political atom bomb he could drop if things happen he doesn't like. However he has such incredible soft power that it's unnecessary, simply quietly voicing his displeasure is enough. He's just made sure no one else can hold that power.

He then gets the senate to assign him to command several provinces that he can command in absentio via legates that he will personally appoint. This isn't unprecedented - Pompeius did the same with Spain. These provinces are the dangerous ones where potential threats to Roman power lie - Germany, Spain, Syria. Coincidentally since those are the hotspots, that's where most of the legions are based. Now the remaining safer provinces he returns to senatorial control - see how respectful he is of Republican virtues? Except since those provinces are safe, they have minimal forces. This means the bulk of the armies are under men he assigns and not ambitious senators. He had learned that who controls an army could be king.

Because he needs the legal authority to command soldiers he is granted imperium, the power normally vested in consuls or praetors to command Roman citizenry in the field. However his is greater than any consuls since it wouldn't do for a situation where a consul could outrank him. Now he has the legal authority to command soldiers regardless if he was elected consul or not (and he was regularly elected until he asked people to choose other candidates).

All of this was in some form or other Republican - he didn't use the dictatorship abused by Sulla and later Caesar. He always participated in senatorial debates and treated them with respect. He never called himself 'emperor' but instead 'first citizen' (princeps). Sure he wielded immense powers but he wasn't a king, he was just another citizen of Rome - just remember he's the first citizen. The term emperor comes from the word imperator which simply was a title given to a successful general by their troops. Win a battle, get acclaimed imperator by your army along with a 10pt popularity boost. Since he's controlling all the armies, he's technically winning all the battles despite another general actually being on field (kind of a 'buck stops here' but in reverse). He had already been acclaimed earlier but his control over the army means that no other senator is getting an opportunity to get that title.