The topic of Byzantine succession always wildly varies with each era of the Empire's history. As thorough as the Byzantine's codification of Roman Law was, there was no formal legal provision or codification regarding succession of the Emperor. So succession took on several forms throughout the centuries and became de facto dynastic in practice for the sake of the stability of an heir being chosen from within the Emperor's family, but this often didn't go smoothly. For example, the longest unbroken Byzantine Dynasty was the Palaiologi, who ruled from 1261-1453, just shy of 200 years.

As for the many ways one could become Emperor because of the absence of legal procedure, here are the most common ways for someone to become Emperor and Autocrat of the Romans:

1. Inherit it from a relative. A family member chosen as a successor was given the title of καῖσαρ (Caesar), the Empire's common title for the heir apparent rather than the Emperor himself during the earlier days of the Roman Empire, who took on the title of βασιλεύς (Basileus) throughout most the Empire's history. While eldest sons of the Emperor were a common option, it was entirely at the discretion of the Emperor as to who he could declare, where nephews of the Emperor (Justin II, nephew of Justinian I who had no children), brothers (Heraklonas, brother of Constantine III), and younger sons (Manuel I, youngest son of John II) should the Emperor deem any other sons or family members as unfit for the title or just simply favor them over other members. This was the most peaceful and preferred methods of succession as the transition of power was most peaceful through a declared successor.

2. Be made Co-Emperor and wait until the Emperor dies (or kill him during the co-emperorship). The appointment of a co-emperor was made for several reasons, but almost never at the initiative of the Emperor himself. The declared heir may still be a minor so a co-emperor would act as regent during his minority. More commonly, however, a co-emperor would be declared as a concession to the Emperor's opposition. Should the nobility plot to overthrow the emperor with a candidate they support, the Emperor would raise the candidate's title to that of his equal to prevent a violent overthrow. Another reason would be due to the age of the Emperor, where he would raise his declared heir-apparent to the title of his equal should he become too old to properly administrate the Empire, an example of this being when Manuel II Palaiologos raised his son John VIII to be co-emperor when Manuel II was at the age of 66, dying 9 years later at 75.

3. Use the army to usurp the Emperor and/or take the throne after his death. Ambitious generals or opportunistic guards to the Emperor would often claim the title with support from the soldiers they commanded to enforce their claim. Most notably Emperor Justin I, Justinian I's uncle, and Emperor Phocas, who led a rebellion that overthrew Emperor Maurice in 602, only to be killed by Heraclius 8 years later, himself a general in the army during the Byzantine-Sassanid War of the time.

4. Political plotting from the Imperial courts and nobility. Members of the nobility would often bribe officials and factions within the imperial court for support, along with outcasting/assassinating rivals. Emperor Nikephoros II Phokas gained power by bribing members of the Imperial court to place him on the throne after the death of Emperor Romanos II. There was a conspiracy that Romanos II was poisoned by factions supporting Nikephoros, but there's nothing solid in terms of evidence.

Of course many of these methods were challenged by other claimants and pretenders to the throne, leading to multiple civil wars and rebellions that would be exhausting to list entirely just to show that even though there were successful attempts to dethrone the Emperor, there were dozens of other failed attempts at doing so.

Sources:

Byzantium: The Decline and Fall by John Julius Norwich

The Oxford History of Byzantium by Cyril Mango

Edit: fixed a mixup of 2 Emperor's names.