There's not a lot more than that to tell.

Plutarch in 67ff gives the narrative; you can also find it in Suetonius, and Cassius Dio.

The day after the assassination Brutus made a speech explaining his actions to the people. The Senate voted Caesar divine honors. But Caesar's will was opened and read publicly, and contained provisions for a vast amount of money to be given to the people. That whipped the people into a mob. The mob burned Caesar's body in the forum. This is an approximation of a public funeral in a sense, and it had precedent in the burning of Clodius' body in the Senate house. THAT symbolism was very clear; Clodius' partisans were literally burning down the aristocracy at Clodius' funeral. Burning Caesar in the forum might be less pointed, but had a similar effect.

After that the mob tried to lynch the conspirators. They didn't get any of them, but they got one guy who shared a name with the conspirators - a certain Cinna - and they ripped him to shreds.

Suetonius in contrast says the will was read in Antony's house. The funeral pyre was set up on the Campus Martius, and instead of a eulogy Antony had the decree of the Senate granting divine honors read. While the body was being carried from the Rostrum (in the forum) to the pyre, mysterious figures set fire to it in the forum. The mob then tried, as in Plutarch, to lynch the assassins and mistakenly killed Cinna.

Dio is much later. He gives whole (invented, to be sure) speeches. Antony addresses the crowd in the forum. This is the closest to Shakespeare, though the speech is very different. After Antony's speech, the mob burned the body in the forum, then they tried to lynch the assassins, and mistakenly killed Cinna.

The facts as reported are not really in question. There's no drastically differing tradition. The differences are in invented speeches and minor details.

I've never seen any scholarly controversy about the funeral. Maybe I'm not just read up on it. But I don't think there's much to say.