You can read the Platonic letters online in English at the Perseus Project. The letters aren't numbered according to any particular chronological or thematic order. If you want to read them in the most probable dramatic order, Debra Nails in The People of Plato, drawing on several different sources, gives the probable dramatic order as: Ninth, Twelfth, Thirteenth, Fifth, First, Second, Eleventh, Tenth, Third, Fourth, Seventh, Eighth, Sixth.
The background for most of the letters is Plato's involvement in the politics of Syracuse, the most important Greek colony in Sicily. Unfortunately, much of our understanding of the events are somewhat tentative, but here is one possible reconstruction. At some point in his travels, Plato visited the city of Syracuse, possibly by invitation and there met the young Dion of Syracuse, who was the brother-in-law of Dionysius, the tyrant of Syracuse. Dion became one of Plato's students. Dion introduced Plato to Dionysius, but Dionysius at some point seems to have taken a dislike to the philosopher and either (depending on the story you follow) tried to assassinate him or to sell him into slavery. Plato left. However, he may have returned after Dionysius's death, when Dion invited him back to educate the new tyrant, Dionysius II. Dionysius the Younger was suspicious of both Dion and Plato, however, and when Dionysius II discovered that Dion was secretly negotiating a peace with Carthage, Dion was banished (or perhaps fled) and toured various Greek cities, becoming very popular in Athens; he may have become good friends with Plato's nephew, Speusippus, who would go on to become the second head of the Academy. Plato tried to negotiate a reconciliation, but failed and left the city; he may have returned a third time for the same purpose, but seems to have been treated almost like a prisoner; he was only able to get away due to the help of Archytas, the philosopher and mathematician. Dion was assassinated several years after this.
This letter is addressed to Dionysius the Younger. Given how short it is, it makes a remarkable number of surprising claims that tell against its authenticity (for instance, that Plato supervised Syracuse for Dionysius II), and doesn't seem to be consistent with the much more likely authentic Seventh Letter, although it shows a good poetic sense. In it Plato says that Dionysius the Younger will die without friends because of his behavior as a tyrant.
This letter is addressed to Perdiccas, the king of Macedon (and elder brother of Philip of Macedon), recommending that he listen to advice from Euphraeus. Although very short, its comment on politics is worth quoting:
Constitutions, like species of animals, have each their own language--democracy one, oligarchy another, and monarchy still another. Many persons would say they know these languages, but for the most part, and with rare exceptions, they fall short of understanding them. The constitution that speaks its own language to gods and men, and suits its actions to its words, always prospers and survives; but it goes to ruin if it imitates another. (321d)
It then ends with a defense of Plato for not speaking up in political matters: He would gladly do so, but when Plato was born, the city was already advanced in age and had bad habits that would make the advice he would give useless.
I confess I find myself a bit puzzled at some of the arguments for this letter's inauthenticity. It has connections with the Republic (493a-b) and the Seventh Letter, without being a blatant case of copying, but these connections are often made the argument for its not being authentic. I am no doubt missing something. Regardless, if it's not by Plato it is a good bit of work in writing from a Platonic perspective.
This letter is addressed to Archytas of Tarentum, in southern Italy; Plato had met him and become friends with him during his travels. It notes a letter that has arrived from Archytas. It is mostly concerned with reports that Archytas is weary of public service; Plato remarks that while it is certainly sweeter to follow one's own interests, there are higher concerns: "none of us is born for himself alone" (358a). One shouldn't refuse public service, especially when considering that doing so would leave public matters in the hands of people with the wrong motivations.
This letter is sometimes panned as too colorless for Plato, but this criticism itself is of doubtful value given that Cicero quotes it twice (De finibus 2.14; De officiis 1.7), and without intending any disrespect to modern classicists, Cicero is a better judge of whether a work was too colorless and commonplace than they are.
This letter is also addressed to Archytas, expressing joy and admiration at having received several treatises from him. Plato remarks that the author of the treatise is worthy of his ancestors, who were said to have descended from the Trojans. Archytas had asked for treatises by Plato in return; Plato says they are not finished, but he is sending copies of them as they are.
There is a note attached to this letter in some manuscripts stating that some have claimed it was not Plato's, thus making it the one letter we definitely know to have had its authenticity questioned even by the ancients. The letter also spells Archytas's name differently than the other letters do.