Speaking entirely legally, there was no usurpation. Following the death of his patron, Basil I became sole Emperor because he had been named co-emperor beforehand, which had been common practice for centuries by that point. Bury's history shows the points leading up to that far better than I could, and there are too many editions to point you at (and mine is electronic, anyway), but I'll give the most important points in the history:

Basil came to Michael III's attention.

Basil murdered Michael's uncle (with, we are told, the Emperor's connivance)

Basil married Michael III's mistress, Eudokia, an Iconoclast with her own fair 'claim' to the Imperial throne via Heraclius, who lost Africa and the Syrias to the Arabs. Not that that meant anything at the time, of course, as I'll explain down the page.

Michael III then adopted Basil as his son. This was not an uncommon event in the Roman Empire and even the Republic beforehand, but by this point I don't believe it had any real expectation of filial relationship.

There was then a falling out between the two men. Edward Gibbon glosses over it as such:

"He supported this unequal association till his influence was fortified by popular esteem. His life was endangered by the caprice of the emperor; and his dignity was profaned by a second colleague, who had rowed in the galleys."

But the historians at the time, and historians since, have generally agreed that Basil suspected that Eudokia's children were in reality the sons of Michael III. This would, of course, put Basil I in an untenable position; having married Eudokia and provided her sons with a legitimate father, they could now be considered legitimate members of the Amorians. Certainly, it was at this point that Basil began to be slighted by Michael III.

The incident that Gibbon mentions is an interesting one. All we today truly know of this man who was allegedly named as co-Emperor is his name, Basilikianos. He has been accused of every crime and of being a disgrace to every station; Gibbon's charge that he was a mere rower in the galleys is the kindest cut. I will say that the balance of probability suggests his family was rather more than that; even under the rule of his erstwhile 'rival', a man who was called his brother served as an official in Constantinople's government. Or so we are told by Bury, sourced from Continuatus.

Regardless, alea iacta est, (or ἀνερρίφθω κύβος, as our principals would no doubt have preferred) the die was cast. Basil's coup certainly reached deep into the Roman government. The eunuch who was on guard wasn't there, and the luckless Basilikianos was. Was he killed? Was he merely blinded or slit across the nose? Was he even permitted to escape? We don't know. No one seemed to think it was important enough to record.

Michael III died, and Basil was crowned. Traditionally, that very night.

And that's where your question comes in, I think. How was this possible? Well, there are two partial answers. First of all, at that time, and for centuries to come, the Imperial dignity was still a trophy for the strong. There was no real 'order of succession' as we understand the term. A general who was popular enough, effective enough, and ambitious enough could see himself on the throne. Michael III's dynasty had risen in such a fashion three Emperors before, and in the distant future so would the famous Alexios Komnenos, whose grandfather Isaac had himself ended the dynasty Basil would go on to found. A mere century and a half after that, a three way civil war between the Komnenii and their rivals would lead to Constantinople being sacked and conquered by the Franks in the so-called 'Fourth Crusade'. Dynastic rule was the ideal, but hardly the norm.

We are told that Basil was popular, especially by Skylitzes. I would be cautious about his work. That there were no rivals present in the city was perhaps the ultimate factor. Unlike old Rome, Emperors truly were made and disposed in Constantinople. By the time generals on the frontier had learned of Basil's succession, it was essentially a fait accompli.

And that brings us to the other half of the equation. Even by the loosely interpreted legal standards of the day, there was no other legal heir to the Empire. Not only was Basil I the strongest leader, and in possession of the city, he had also been named Co-Emperor. Admittedly, he had then murdered his benefactor, but as discussed that was hardly the, aheh, faux pas we would see it as today.

Also to be considered...

Michael III was hardly the failure that Gibbon viewed him as. But he wasn't much of anything. His personal rule is marked more by lack of diligence than by great feats. Great feats were done in his name, of course, but even then people knew about taking credit for the work of others. His greatest feats were on the battlefield, but he ultimately wasn't given the chance to make a real mark for himself in the rule of the city, and he knew it and resented it. Ultimately, that may have led to his murder of his uncle, who apparently was a popular figure and an effective one. And fairly or not, he took the blame; the idea that Basil instigated it seems a bit pat, but that's what's come down to us.

Was he a drunkard? Honestly, probably. Even with the best will in the world, Michael III is too consistently charged with being a heavy drinker for it to be completely unjustified. And it makes some sense, considering the position Michael found himself in. He was Emperor for almost his entire life and yet had never been properly trained or taken seriously; he had had to exile his own mother in order to end the regency, and then he fell into the hands of his uncle.

You had a drunk who had, in memory of most of his government, overthrown his own regents and ordered the murder of his own uncle while preparing for a military campaign. On the other hand, you had a ruthless political figure who'd pulled himself up to the Empire's heights by his own will and demonstrated that he was very dangerous to cross.

It was only ever a matter of when.