Ok, short version is there was a rumor that he had a relationship with Nicomedes of Bithynia, and he was famous for having affairs, but this thread is a pretty sensational take.

Claim 1) Contemporary Roman writers slutshamed Julius Caesar

There were apparently a lot of contemporaries alluding to this affair, but they are reported by Suetonius, writing in the early 2nd century, so... 150ish years later? He has a lot of good information; he was Hadrian's secretary and had access to the Julian archive and used it, but he also passes along rumors. Here's what he says:

There was no stain on his reputation for chastity except his intimacy with King Nicomedes, but that was a deep and lasting reproach, which laid him open to insults from every quarter. I say nothing of the notorious lines of Licinius Calvus:

"Whate'er Bithynia had, and Caesar's paramour."

I pass over, too, the invectives of Dolabella and the elder Curio, in which Dolabella calls him "the queen's rival, the inner partner of the royal couch," and Curio, "the brothel of Nicomedes and the stew of Bithynia." I take no account of the edicts of Bibulus, in which he posted his colleague as "the queen of Bithynia," saying that "of yore he was enamoured of a king, but now of a king's estate." At this same time, so Marcus Brutus declares, one Octavius, a man whose disordered mind made him somewhat free with his tongue, after saluting Pompey as "king" in a crowded assembly, greeted Caesar as "queen." But Gaius Memmius makes the direct charge that he acted as cup-bearer to Nicomedes with the rest of his wantons at a large dinner-party, and that among the guests were some merchants from Rome, whose names Memmius gives. Cicero, indeed, is not content with having written in sundry letters that Caesar was led by the king's attendants to the royal apartments, that he lay on a golden couch arrayed in purple, and that the virginity of this son of Venus was lost in Bithynia; but when Caesar was once addressing the senate in defence of Nysa, daughter of Nicomedes, and was enumerating his obligations to the king, Cicero cried: "No more of that, pray, for it is well known what he gave you, and what you gave him in turn."

These look like direct quotes. We have edicts of Bibulus, letters from Cicero, Osgood thinks the invective of Dolabella was a speech. So contemporary writers seems fair.

Claim 2) Caesar was a thirsty, thirsty bottom.

Osgood did an analysis of these charges and similar charges brought against Caelius. Amy Richlin found that these sorts of charges and other charges of sexual "misconduct" could be usefully deployed against young politicians, and Osgood judges these to fall into that category. This sort of invective doesn't represent Caesar's actual preferences.

Claim 3) Suetonius: I heard that Caesar was a big slut and also he liked buttsex and oral.

We already saw Suetonius go through his list.

Claim 4) Cicero, to the gathered senate: Caesar takes it up the ass.

That refers to this bit of Suetonius:

Caesar was once addressing the senate in defence of Nysa, daughter of Nicomedes, and was enumerating his obligations to the king, Cicero cried: "No more of that, pray, for it is well known what he gave you, and what you gave him in turn."

Claim 5) The sex was so good that when Nicomedes died he lift his entire kingdom to Rome.

Nicomedes did leave his kingdom to Rome, but that was a strategy Hellenistic kings had practiced since Attalus left Pergamon to Rome in 133. We've already seen this kind of invective doesn't have to be grounded in fact, so it probably wasn't because of the sex.

Claim 6) Julius Caesar was stabbed for being a bottom.

No.

Claim 7) There was a song his soldiers would sing when they marched announcing all this to the whole world.

There was a tradition of soldiers singing satirical songs about the general during a triumph, and indeed Caesar's troops sang such a song, and Suetonius saved it for us:

"All the Gauls did Caesar vanquish, Nicomedes vanquished him;

Lo! now Caesar rides in triumph, victor over all the Gauls,

Nicomedes does not triumph, who subdued the conqueror."

Claim 8) Caesar has been dead for 2062 years.

That is true now, but wasn't 8 months ago, and won't be in 6 months in the future.