Antony was not only at Caesar's funeral, he was sort of the main event, so to speak. Antony's funeral oration for Caesar is one of, perhaps the, defining moments of his career. Antony's funeral oration caused a mass riot in the city, which was so terrifying that it drove Caesar's assassins out of the city. This was a serious and sudden change of heart on the part of Antony, who had been quite moderate with the murderers earlier. Immediately following Caesar's murder the city was in a state of near-anarchy for several days, as the murderers barricaded themselves in on the Capitoline and the Caesarians recovered their strength. Antony as sole consul, despite having the troops to do so, commanded by Caesar's magister equitum Marcus Lepidus, did not assault the position on the Capitoline, though it was held, says Appian, only by gladiators, but instead posted watches throughout the city to maintain order. On March 17 the senate convened in the Temple of Tellus and Antony delivered an inspired oration, convincing the senate to pardon the murderers and allow them the provincial commands that Caesar had intended for them, on the condition that Caesar's legislation be considered inviolate. It was a brilliant political move, reconciling the opposing sides and preventing another outbreak of civil war, as well as cementing Antony's place as the de facto head of the state. Though Antony had accepted Dolabella (his enemy, who had made a disaster of Antony's governorship of Italy in 47), who had been promised the consulship by Caesar (though he was only 25 and under the minimum age for the office), as his consular colleague, he was very much in control--Dolabella had defected to Antony from Cassius, and Antony believed that he was to be named Caesar's legal heir in Caesar's will. On March 19 the will was read aloud, and Antony discovered to his horror that he was not in fact Caesar's legal heir, but was hardly named at all! Caesar's funeral was held publicly the next day, and Antony, as consul, delivered the oration, inciting a riot and driving the tyrannicides from the city--a total and sudden change of heart that, in point of fact, confirmed Antony's power, driving out and rallying the state against a perceived enemy just as the legitimacy of Antony's claim to Caesarian leadership might be thrown into question.

That said, we have no transcript of what was said, and Shakespeare's words are an invention based loosely on what Plutarch reports. Plutarch says that Antony gave an inflamed speech, waving Caesar's bloodied toga and speaking of the great injustice of the murder of such a great man. Suetonius reports that Antony did not really give much of an oration at all, instead reading the edict that named Caesar a god and delivering only a few remarks of his own. In the life of Brutus Plutarch adds that at the funeral Antony had Caesar's will read aloud, and that Caesar's body would be displayed to the public, a decision that Plutarch says Brutus agreed to, despite Cassius' disapproval. The contents of Caesar's will, which kindly provided for 75 drachmas to be distributed to every citizen, and the sight of his mangled body proved too much for the people of the city, and urged on by Antony's words they grabbed torches and stormed off to set fire to the conspirators' houses. The fullest account of the funeral oration is Appian's, although I do not believe that Shakespeare had access to it (and he read Plutarch in translation, not Greek). Appian reports that first Caesar's will was read aloud, and the people were greatly disturbed by the fact that Caesar, whom the conspirators had denounced, loved the state so much as to contribute enormous sums of money towards public works and to each citizen--they were still more agitated when Antony read that Decimus Brutus, one of the murderers, was to be named Caesar's alternate heir after Octavian. Caesar's body was brought in by his father-in-law Piso, amid great lamentation. Antony then read aloud the various honors granted to Caesar, emphasizing certain ones beyond others (as would have been not unusual in Antony's Asiatic orational style), and in particular pointing out Caesar's famous clemency, which many of the murderers had taken advantage of. Antony next read aloud the oaths taken by various people to protect Caesar and stressed that the state now stood once again on the brink of civil war, thanks to the murder. Finally Antony accepted Caesar as a god and listed his many victories, before in a frenzy tearing off Caesar's bloodied toga and waving it before the crowd. Both Appian and Suetonius report that at this point Antony quoted Pacuvius, although only Suetonius gives the Latin (Appian translates the line, which was originally written in Latin, into Greek):

men servasse, ut essent qui me perderent?

Did I save them, that they would be the ones to kill me?

At this, says Appian, the crowd flipped out and started seizing torches, intent on burning the houses of the assassins (Appian also mistakenly reports that they burned the Curia, conflating the riots that occurred after Clodius' murder with those at Caesar's funeral).

As for the famous "Friends, Romans, Countrymen" line, it seems to be an invention of Shakespeare's. Certainly Antony is not reported by any of our sources to have asked the crowd to "lend their ears" or anything of that sort. The address of "Friends, Romans, Countrymen" has a little bit of a resemblance to Appian's repetition of πολῖται (citizens) at several points in the speech. Appian's report is the only one that actually presents an oration, Plutarch's various reports simply say that "yeah, Antony said some stuff about this and that," but doesn't actually try to tell us Antony's words. Appian's oration begins:

Οὐκ ἄξιον, ὦ πολῖται, τοσοῦδε ἀνδρὸς ἐπιτάφιον ἔπαινον παρ᾽ ἐμοῦ μᾶλλον, ἑνὸς ὄντος, ἢ παρὰ τῆς πατρίδος ὅλης αὐτῷ γενέσθαι

It is not right, oh citizens, that the funeral oration of praise of such a great man be delivered by me, a single man, than by the entirety of his country.

Not very much like Shakespeare, and as I said I don't believe Shakespeare knew about Appian. More likely Shakespeare is trying to adapt the familiar Roman term Quirites (which, presumably, is what Appian means when he says πολῖται, a word that Antony surely would not have used), the general term used to address the citizen body as a whole