You can read Axiochus online in George Burges's translation.
(in order of appearance)
The son of Axiochus. He is also a character in Plato's Euthydemus.
Axiochus was the uncle of Alcibiades; he was also a relative of Aspasia, the companion of Pericles. He was closely associated with his nephew, and had to flee Athens when, like Alcibiades, he was indicted for defamation of the statues of Hermes. He returned to Athens late in the War. His last known act was defending the generals who were at the battle of Arginusae, which is mentioned in the dialogue.
In addition there are two characters who are explicitly present, at least at the beginning of the dialogue, but who do not speak at all:
Damon the musician, Clinias's music teacher; he was a student of Prodicus, and advisor of Pericles, and is mentioned occasionally in the Republic and Laches.
Charmides, who is Plato's uncle and eventually became one of the Thirty Tyrants; he is, of course, a character in Charmides.
Socrates, who narrates, is walking to the Cynosarges when he hears someone shouting his name. It turns out to be Clinias, who is walking with Damon and Charmides. Clinias has tears in his eyes, and asks Socrates to come visit his father, who has been unwell and is miserable because of his impending death. He asks Socrates to comfort him. They hurry to Axiochus's house. There they find Axiochus weeping and groaning. Socrates starts the discussion:
"Axiochus, what's all this? Where's your former self-confidence, and your constant praise of manly virtues, and that unshakable courage of yours? You're like a feeble athlete who put on a brave show in training exercises and lost the actual contest! Consider who you are--a man of such an advanced age, who listens to reason, and, if nothing else, an Athenian!--don't you realize that life is a kind of sojourn in a foreign land (indeed, that's a commonplace, on everybody's lips), and that those who have led a decent life should go to meet their fate cheerfully, almost singing a paean of praise? Being so faint-hearted and unwilling to be torn from life is childish and inappropriate for someone old enough to think for himself." (365a-b)
Axiochus agrees, but says that now that he's close to death, all the powerful arguments lose their strength and fear assaults his mind, because he will lose the good things of life and will become food for the maggots.
Socrates suggests that he is contradicting himself: he is upset both that he is losing sensations and that he will feel the loss. Just as nothing about what happened before he was born should upset him, so nothing about what happens after his death should harm him. He then suggests that the body is a prison and that being released from it is a good. Axiochus asks him, if he thinks living is bad, why he continues to live, but Socrates replies that he is not an expert on this subject, and so his remarks are taken from Prodicus, who recently gave a performance in which he denounced living. Socrates tells Axiochus what he remembers of the presentation. He ends by talking about the futility of trades and professions, including Axiochus's own, and gives as an example the Aginusae case.
Toward the end of the Peloponnesian War, an Athenian fleet under eight generals defeated a Spartan fleet near the islands of Aginusae; the Athenian fleet was not in great shape, so the victory was very unexpected, and nobody was prepared for it. In the course of the battle, however, twenty-five Athenian ships were sunk or sinking; rescue attempts were made, but a storm coming at the end of the battle made rescue very difficult. Further, the generals had what seemed like a chance to destroy the fleeing Spartan fleet, and the generals decided to devote of their resources to that (they failed, in fact, to destroy the Spartan fleet). When the Athenian assembly heard about the generals leaving so many Athenians to drown in the sea, they were furious. They deposed the generals and demanded that they return to Athens to stand trial. Two of the generals fled, but six returned.
When they did, some in the assembly suggested that the assembly should just vote on their innocence or guilt directly. Euryptolemus (a cousin of Alcibiades) and Axiochus opposed this as illegal, but they withdrew their opposition when people started proposing motions to apply the same treatment to them. As it happened, the presiding officer chosen by lot for that day was Socrates, and Socrates refused to put the proposal to a vote because it was illegal. Euryptolemus put forward a resolution to try each general separately in a proper trial, and it passed, but Theramenes and others managed to repeal the vote and the next day got the original proposal put forward by the new president of the assembly, which passed. All six generals were voted guilty and executed. So politics, Socrates notes, is not a very pleasant trade. Axiochus agrees and says that he essentially has given up politics ever since that incident, and Socrates continues with Prodicus's arguments. Axiochus replies, however, that they are all just words, and that an argument in order to be of any value would have to come down to his level.
Socrates goes on to present some arguments for the immortality of the soul, and gives an account of the afterlife, including the Elysian Fields and Tartarus, which he attributes to Gorbryas the Persian. These arguments do convince Axiochus, so that he almost longs for death. He says that he would like to be alone to think over what Socrates has said, but he asks that Socrates come back later in the day. Socrates promises and says that he is going to go back to his walk to the Cynosarges.
* The Cynosarges were a gymnasium near Athens where people went for all sorts of events. Quite a few philosophers and sophists seem to have taught in the area.
* Since Socrates sarcastically remarks that Prodicus teaches nobody for free, it seems clear enough that he should not be taken as endorsing anything he attributes to Prodicus.
* The Athenian assembly soon regretted its decision to put the generals from Arginusae to death, and decided to make it better by bringing charges against those who had advocated for the vote; those men fled Athens.
* Gaubaruva, or Gobryas, was a very common name among Persians, so I'm not sure which one is in view here.
A dialogue of this sort is tricky to interpret, but I think a plausible way to read it is to take Socrates as initiating conversation with irony, in order to provoke Axiochus into thinking about the arguments rather than his own feelings. He doesn't seem to expect Axiochus to agree to the common views or the views of Prodicus, but only to respond to them. He then builds on Axiochus's response to argue for the immortality of the soul:
"As well as many other fine arguments for the immortality of the soul, a mortal nature would surely not have risen to such lofty accomplishments that it disdains the physical superiority of wild animals, traverses the seas, builds cities, establishes governments, and looks up at the heavens and sees the revolutions of the stars, the courses of sun and moon, their risings and settings, their eclipses and swift restorations, the twin equinoxes and solstices and Plaieds storms, summer winds, torrential downpours, and the violent course of tornadoes, and establishes for all eternity a calendar of the states of the universe, unless there really were some divine spirit in the soul which gives it comprehension and insight into such vast subjects...." (370b-c)
On the basis of this Socrates argues that Axiochus is passing away not into death but into immortality, in which he will be free from the entanglement of the body and be able to devote life to study of nature and practice of philosophy "in the bountiful midst of Truth" (370d). Giving an afterlife account is a fairly standard Socratic gambit after this kind of argument.
However, I suspect that there is something more here, although it is difficult to pin down exactly what it is. Axiochus's defense of the generals is made fairly prominent in the middle of the dialogue; civilized life is part of the brief argument for the immortality of the soul; and the afterlife myth, as afterlife myths in the genuine Platonic myths often are, seems devoted to emphasizing the importance of virtuous life, which the ancient world saw as closely connected to political life. And Socrates is quite clear about the overall moral of the story:
"...I am moved by argument, and I know only this for sure: every soul is immortal, and also, when removed from this place, free from pain. So whether above or below, Axiochus, you ought to be happy, if you have lived piously." (372a)
Moreover, Socrates seems to have aimed at exactly what he achieved; he now has Axiochus thinking not about pain and loss of pleasure, but about virtuous life.
Quotations from Jackson P. Hershbell's translation in Plato, Complete Works, Cooper and Hutchinson, eds., pp. 1734-1741.