What did Megas Alexandros and Julius Caesar actually sound like?

Although it's possible to point to this or that reconstruction of the pronunciation of Greek and Latin, none of them are in any way satisfactory. We have very little idea of what a native speaker of any Greek dialect or any form of Latin prior to late Medieval Latin might have sounded like.

We'll start with Greek and Alexander (incidentally, while Megas Alexandros is a technically correct form, in reality it's baby speech. In Classical and Hellenistic forms of Greek the final s in megas would have consistently been elided with the a in Alexander, to form Megalexandros. This, however, is not the usual form of the title, which is usually represented in Greek texts as Alexandros ho Megas). Greek is extremely difficult to reconstruct. In the case of both Greek and Latin the currently accepted pronunciations, which are taught in Latin and Greek courses around the world, are slightly updated versions of those proposed by Sturtevant during the mid-20th Century. Although it's useful for understanding more or less how to pronounce words, it's unsatisfactory for reconstructing an ancient accent. Greek is notoriously problematic for several reasons. First of all, there were several different forms of Greek, which would be spoken in different places at different times. Each dialect was pronounced differently, and Greek pronunciation varied greatly between periods. For example, Homeric Greek probably had this sort of nasally, almost whiney sound (a teacher of mine liked to say that if you sound annoying you're doing it right), as opposed to Hellenistic Greek, where the vowels had migrated down into the palette. There are also several issues with our understanding of the Greek language. Unlike with Latin, where there is a fair amount of documentation from the early Middle Ages describing forms of the language and features thereof, the Greeks never found it necessary to do this and by the Middle Ages when such things are recorded the language had changed enough that it is no longer Ancient Greek, but an early form of Modern Greek (i.e. Byzantine Greek). As a result we know surprisingly little about how Ancient Greek worked. This also throws off our understanding of the pronunciation--until Sturtevant proved that the pronunciation found in medieval handbooks couldn't possibly be correct we thought that Classical Attic was pronounced the same way as was laid out in the Renaissance, where they assumed that it was pronounced the same way as in Byzantine texts. Supremely important is the question of vowel quantities and the accent. We have a decent idea of the vowel qualities, but we can't be totally sure of vowel quantities, i.e. how long the vowels were. We assume that long vowels were simply twice as long as the short ones, but it's very difficult to tell what that might have sounded like, since there are very few parallels in surviving languages. Of extreme confusion is the accent. English had a stress accent, as did Latin, but did Greek? We actually have no idea and it's intensely debated. If Greek had a stress accent (and it's likely that very early forms of Greek did--Macedonian and Epirote Greek may have preserved it well into the historical period) it disappeared sometime during the Hellenistic Period an was a vague memory by the early Middle Ages. Some dialects may have never had a stress accent to begin with, such as Ionic (which also seems never to have had a real aspirate, except maybe during the Dark Age) whereas others like Boeotian may have preserved it (if it existed). Whether Classical Attic had one is totally unknown to us, but we know that Greek poetry shows no signs whatsoever of the existence of a stress. We do know that Greek had a musical, or tonal accent, in which different syllables would have differing pitches. The only real language that preserves this is Chinese, which has a lot more pitch variations than Greek would've. Greek had three main accent-marks (only used in Hellenistic texts, since prior to that it was assumed a reader knew Greek well enough to figure out what the hell it said without the need for accent-marks)--the acute, circumflex, and grave. These indicate rising and falling pitches, but there's a problem here. We can make good guesses of what exactly they represent, but we don't know for certain--for example, we have no actual evidence for what the grave is, and just assume that it represents a falling pitch. Moreover, we don't have any idea how much the pitches changed or how high or low it went. No clue.

So the short version there is that we have no idea. However, we do know some things about how Macedonian Greek differed from Classical Greek. Alexander and the Macedonian nobles spoke the koine in public. The koine was an artificial dialect based mainly on Attic and Ionic that was used as the lingua franca (koine means "common thing") after about the middle of the 4th Century. We don't have the first idea how it was supposed to sound, and it's almost certain that people from different regions would've pronounced it incredibly differently and almost unintelligibly (there are many Greek texts making fun of foreigners' pronunciation of koine). What we surmise is that since it was totally artificial it may have had a very haughty sound to it. Among themselves the Macedonian nobles would've spoken the Macedonian dialect of Greek, which Alexander seems to have avoided often. The Macedonian dialect was extremely backward with respect to the rest of Greek. It preserved sounds and words that had died out in the early Archaic Period, and was made fun of for sounding like a barbarian speech.

So what about Caesar? Latin is slightly easier to reconstruct, but once again we can't know the details which would be crucial to being able to distinguish between a basic understanding of the pronunciation and a native-speaker's (or even a foreigner's). Latin is particularly difficult because the language changed rapidly when the Romans encountered Greeks in large numbers, a change that only was really completed until just before Caesar's time. For example, Latin seems to have had a stress accent, but no musical accent. Early Archaic Latin poetry in the Saturnian meter preserves the stress, but after the contacts with the Greeks, and particularly after Ennius' grecophile poetry the stress accent disappears from Latin verse and it is replaced by Greek meters, which measure feet according to the lengths of syllables. The stress accent seems to have become weaker and weaker until it disappeared some time during the Late Principate or the 3rd Century, A.D. We're really very unsure of how strong the stress accent was during Caesar's time or really any other. Another thing that's important is that there was a syllabic change just before Caesar's birth, which is first really attested in Cicero. Inflectional endings which in Archaic Latin would've been pronounced on way suddenly changed. For example, the nominative singular of tuus, "yours" is taught as tuus in Latin classes today. But until around 100, B.C. it would've been pronounced, and was written, tuos (which is confusing because it's the same as the masculine accusative plural of the same word, but anyway...). Cicero represents a major change in the Latin speech which probably occurred before, but which was not used in public speeches until he did it. There was also, of course, Vulgar Latin, the common speech used in the streets. One thing that's usually misunderstood is the way that Vulgar Latin worked. Movies and plays seem to think that the lower classes spoke a different form of Latin than the upper classes, rather the way that Victorian gentlemen spoke the King's Speech, but the lower classes spoke an early form of Cockney. This isn't true (at least until the Late Empire, when the language started to branch out). All Romans would've spoken Vulgar Latin in the streets or at home regardless of class and would've spoken Classical Latin on public occasions and so forth. Vulgar Latin is not the same as slang. It's a separate, simplified form of the language that was easier to learn and faster to speak. It invades proper Latin speech quite often, particularly in Catullus and other epigrammatic poets, but also in Cicero (who, despite his reputation for being stuffy and archaic among students learning Latin, was actually quite innovative and renowned for introducing slang and popular forms of words and grammar into his speeches and letters). With Cicero we can almost sort of kind of maybe construct a decent idea in our minds of how the man spoke, since we have a lot of his private letters and speeches and can get some idea of what phrases and expressions he might have been fond of. With Caesar we can't know. The only writings by Caesar we have, apart from a couple of very official letters to Cicero, is his commentaries, which being dispatches to the senate on his military activities are absurdly official and extremely artificial. The man did not speak like that, and he couldn't have spoken like that. On thing that Roman nobles often did, however, was avoid speaking Latin and speak Greek. This was rather frowned upon in some circles, such as Cicero's, and it was particularly unpopular in Caesar's time but instead gained importance during about the middle of the Principate, when nobles and intellectuals used Greek to distinguish themselves from the growing "mob." Caesar certainly knew Greek (he was taught rhetoric at Rhodes and every good Roman noble since Scipio Africanus made it the norm was taught Greek as a boy) and Suetonius claims that Caesar shouted kai su, teknon as he was being stabbed ("You, too, my child?!" The Latin, "et tu, Brute?" is first attested in Shakespeare) but there's no reason to believe that he would've spoken Greek regularly, particularly since it was rather unpopular to do so until much later.

The short answer? We have no idea and we can't know.