Interesting stuff, thanks.
1. By whom was this charge raised?
This isn't clear. The royal family had asked that he be brought to justice.
2. Whose prisoner was he?
Initially he was captured by Ruffo's royalist Neapolitan troops. On 27 June, Sir William Hamilton wrote to Ruffo:
“My Lord [Nelson] begs me to add, that if your Eminence judges it expedient to send Caracciolo and the rest of the other rebels to him, according to his proposal yesterday, he will dispose of them".
So he became Nelson's. Later that day, Hamilton added:
“... Caracciolo and 12 of the most infamous rebels are this day to be sent to Lord Nelson. If my opinion is relished they should be sent directly to be tried by the judge at Procida, and such as are condemned be brought back and executed here. Caracciolo will probably be seen hanging at the yardarm of the Minerva, Neapolitan frigate, from daybreak to sunset, for such an example is necessary for the future marine service of his Sicilian Majesty, and particularly as Jacobinism had prevailed so much in the Neapolitan marine.”
He was brought onboard Nelson's ship on 29 June.
3. The form of the court
As Hamilton mentioned, there was a court operating on the island of Procida, where a royal judge (Speciale) had been processing prisoners since April (working with Troubridge). Royal judges were also active in Naples as on 15 June Ruffo had established a court under Bisogni to try rebels.
So it seems to me that a Board of Naval Officers (five Neapolitan officers) reported it to be a capital offence, but then Nelson took it upon himself to confirm that and specify the means of execution. There was no appeal to the king, nor any use made of the Neapolitan judiciary. This seems a strange approach as other Neapolitan naval officers were brought to trial before a giunta composed of generals in July 1799.
Hamilton concludes, but calls the Board of Naval Officers a court-martial:
“My dear Sir, — I have just time to add Caracciolo has been condemned by the majority of the court-martial, and Lord Nelson has ordered him for execution this afternoon at 5 o'clock, at the foremast yard arm of the Minerva, and his body thrown into the sea. Thurn represented it was usual to give 24 hours for the care of the soul. Lord Nelson’s orders remain the same, although I wished to acquiesce with Thurns opinion. All is for the best. The other criminals will remain at the mercy of his Sicilian Majesty on board the polaccas — in the midst of our fleet. Lord Nelsons manner of acting must be as his conscience and honour dictate, and I believe his determination will be found best at last."
It seems a strange sort of jurisdiction has been applied, but, of course, what is more unusual is that his likely fate and sentence was clear two days before he arrived on Nelson's ship.