Our belief that Caesar suffered from epileptic fits is almost entirely due to a couple references in Plutarch. Plutarch mentions that Caesar had poor health, and refers to "fits" or "attacks" (the Greek could mean either) of his disease. Plutarch also describes that Caesar supposedly had an attack of his disease on the floor of the senate, where he cried out in a loud voice. But Plutarch never describes the symptoms of the disease and the closest thing we have to a description is his account of Caesar's actions at the senate, where he supposedly cried out uncontrollably that anyone who wished to kill him could. Plutarch says that Caesar later claimed that it was due to an attack of his disease, and that people afflicted with that disease (note that Plutarch never refers to it as the sacred disease or by any of the usual terms with which epilepsy was described--it's always just "the disease") tend to become giddy when talking before large crowds, which certainly sounds more like stage fright than epilepsy. Suetonius, however, explicitly says that Caesar was attacked by the "electoral sickness" (morbus comitialis, the usual Latin term for epilepsy, so called because epileptic fits were considered due grounds to postpone elections) twice on campaign. But Suetonius doesn't describe the attacks at all.

Now, although that's rather thin evidence, usually when you've got two sources that agree on something like this and there's no good reason to doubt it you'd accept it. With Caesar, there is a good reason to doubt it. In antiquity epilepsy was not seen, at least in the Mediterranean, as some sort of curse (usually). Referred to as the "sacred disease" among the Greeks, it was generally perceived by those uneducated in medicine or natural philosophy as being possession by a deity, although whether malicious or not was usually up to the specific case. The Greeks and the Romans knew the causes and symptoms of epilepsy quite well since at least the Hippocratic treatise On the Sacred Disease, the first medical treatise which seriously attempts to systematically disprove the idea that disease is caused by supernatural causes and prove that it is due to physical causes that can be targeted by regimens. Nevertheless, even though anyone educated knew better, epilepsy still stuck in the popular imagination, which frequently shows through in literature. Epileptics were often the possessors of great powers, being the subjects of divine favor, or were great men just generally favored by the gods. Generals in particular seem to have been described as having epilepsy. Alexander, Caesar, and (if memory serves) Hannibal are all described by sources as having epileptic fits. This makes it seem suspect. Plutarch's testimony is particularly suspect, not only because he never calls it epilepsy and the symptoms don't match up, but also because Plutarch pairs Caesar's biography with Alexander's and seeks to draw similarities between the two. Alexander is mentioned as having epilepsy in several sources (whether he actually did or not is a question for another day) and with this in mind it's not surprising that epilepsy pops up with Caesar as well. Suetonius is harder to refute. Keep in mind, however, how Suetonius was getting material--essentially through gossip, as he himself admits readily. While suetonius should not be dismissed quite so readily as some on this subreddit would like to think, critics of the theory that Caesar was epileptic note that all kinds of rumors are spread in an camps about generals, and frequently Suetonius takes these rumors seriously, or at least decides to write about them.

Now, could Caesar have had some similar disease that was misidentified as epilepsy? Doubtful. Like I said, the Greeks and Romans knew epilepsy pretty damn well, and it was one thing physicians loomed out for constantly. Modern retrospective diagnoses aren't really that helpful, since diseases were described very differently in antiquity, making such diagnoses more or less speculation (it also doesn't help that many of the people doing these diagnoses frequently don't know the language of the texts and can't perform the kind of close reading and analysis that would be necessary to really understand the descriptions of symptoms). And in the long run it doesn't really matter much--what difference does it make exactly what was wrong with Claudius, for example. We have all kinds of (frequently conflicting) descriptions of his disease, many of them pure gossip and many nothing more than slander, and picking through such things is not accurate nor does it help much, distracting us from the point--that people at least believed+ that there was something g wrong with Claudius, which affected his rule. Much the same thing can be said of Caesar

I'll edit for a source later as the biography I'm reading is at home, but I just happened to read this chapter about Julius Caesar last night so it's still fresh in my mind! At the time, medical knowledge about a condition such as epilepsy was limited and so it's possible that the average person would have considered an episode to be supernatural in nature. For example, less than a century after Caesar's prime, it is written in the Bible that Jesus cured epilepsy from a young boy by expelling an evil spirit:

"But even while the boy was on his way, the spirit hurled him to the ground in a dreadful convulsion. Then Jesus reprimanded the evil spirit, healed the lad and handed him back to his father. And everybody present was amazed at this demonstration of the power of God." - Luke 9:42-43

The biography I'm reading did not mention whether Julius Caesar had any epileptic episodes while before the Senate or other dignitaries, or out in public, which is surprising given how many public speeches he gave and how often he attended Senate hearings. One would think that his enemies, of which there were many, would have used knowledge of his condition and Roman superstitions to their advantage, given that they attacked Caesar time and again (e.g. spreading rumors of his affair with Nicomedes, producing fake letters under Caesar's name tying him to conspiracies, trying to block his legislation by declaring every day during his year as Consul to be a holy day, etc). The biography did state that he sought to overcome his illness by partaking in the difficulties of the common soldier during military campaigns and eating simple foods, sleeping out in the open, marching, and other hardships intended to cast out what he considered to be a weakness, so to speak. This served not only to endear him to his soldiers (along with fighting with them in the front lines), but seemed to improve his condition as well.

Mods, please leave this post up and I will edit in the source when I return home in a few hours.

EDIT: from Julius Caesar by Philip Freeman, page 94:

"Caesar struggled with poor health, regularly suffering crippling headaches and recurring bouts of epilepsy. The ancients knew this latter condition as the falling sickness or sacred disease, because victims would collapse to the ground in seizures that seemed to onlookers like divine or even demonic possession. Epilepsy was well known and widely discussed in the ancient world, especially in Greek medical texts. Even in the gospels, Jesus heals a severely afflicted epileptic boy who is believed to be possessed by an evil spirit. Caesar's epilepsy did not strike him until adulthood, and then it seemed to have been particularly frequent during military campaigns. But as Plutarch says:

'He never allowed his weakened health to slow him down, but instead used the life of a soldier as therapy. He marched endlessly, ate simple food, slept outside, and endured every hardship. In this way, he strengthened his body against illness.'"