I was wondering how the Greeks in the archaic or classical age wrote letters, if there was some sort of convention for them, thus I searched for Ancient Greek letters but found nothing.
Is somebody aware of anything about it?
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There are, plenty. This article contains a very comprehensive reference lists regarding Greek Epistolary research, including references to letters and collections of letters. Unfortunately, you need a subscription to see the full content. I share the key paragraph, with the mentioned references:
Texts and Commentaries
There are two excellent selective anthologies of literary letters: Costa 2001 includes both fictive and pseudohistorical letters in the text with commentary, while Trapp 2003 offers a wide selection of free-standing and embedded letters, as well as a succinct definition of “letter.” Both books are excellent starting points for getting a sense of the sheer variety of epistolary works. The most complete collection of Greek epistolary texts, however, remains Hercher 1873, with its facing Latin translation. Some letter collections have been published in standard editions, such as those of Teubner, Budé, and Loeb—Benner and Fobes 1949 and Jones 2006 are examples of these—but coverage is spotty. The pseudonymous letters of Euripides exist in the scholarly edition Gösswein 1975; the epistolary novel Chion of Heraclea is well served in Malosse 2004 and Aristaenetus appears in Drago 2007. However other epistolary texts remain understudied and difficult to find outside Hercher 1873.
Benner, A. R., and F. H. Fobes, eds. 1949. Letters of Alciphron, Aelian, and Philostratus. Loeb Classical Library. Cambridge, MA: Harvard Univ. Press.
Three imperial authors of fictive letter collections published in a single Loeb volume, with Greek text, literal translations, and limited notes.
Costa, C. D. N., ed. 2001. Greek fictional letters: A selection with introduction, translation and commentary. Oxford: Oxford Univ. Press.
A selection of fictive epistles (Aelian, Alciphron, Philostratus, Aristaenetus), and epistles allegedly by historical figures (Anacharsis, Diogenes, Crates, Socrates, Aeschines, Hippocrates, Themistocles, Euripides, Chion); includes Greek text, facing translation, and brief commentary.
Drago, A. T., ed. 2007. Aristeneto: Lettere d’amore. Lecce, Italy: Pensa Multimedia.
A lengthy (70 pp.) introduction covers questions of manuscript history, and is followed by the Greek text of Aristaenetus with an Italian translation, and extensive commentary focusing on source texts and parallel passages.
Gösswein, H. -U. 1975. Die Briefe des Euripides. Meisenheim am Glan, Germany: Anton Hain.
First edition since Hercher 1873 of the five letters attributed to Euripides. The author consulted all thirty-four available manuscripts, and prints an introduction discussing Authenticity and date of composition, a critical text with a German translation, and a commentary.
Hercher, R., ed. 1873. Epistolographi Graeci. Paris: A. F. Didot.
A massive (over 800 pages) work of late 19th-century scholarship, organized alphabetically by author, with critical notes, a section on Epistolary Theory, Greek text with facing Latin translation, and useful indexes. Still the most comprehensive edition of epistolary texts availble, and the source for many of the Thesaurus Linguae Graecae (digital library) texts.
Jones, C. P., ed. 2006. Letters of Apollonius; Ancient testimonia; Eusebius’s reply to Hierocles. Cambridge, MA: Harvard Univ. Press.
A scholarly edition of the ninety-one letters atrributed to Apollonius, along with judicious remarks on authenticity, organization, manuscript transmission, and historical context.
Malosse, P. -L. 2004. Lettres de Chion d’Héraclée. Salerno, Italy: Helios.
Greek text and facing French translation, introduction, and notes. Investigates issues of genre, readership, and historical context. Suggests a later date (4th century CE) than previously assumed.
Trapp, M., ed. 2003. Greek and Latin letters: An anthology with translation. Cambridge, UK: Cambridge Univ. Press.
An anthology of seventy-eight Greek and Latin letters, including facing translation and commentary. Opens with a succinct and now widely accepted definition of the letter. Selections are organized according to private, public/official, embedded, and Epistolary Theory, and they highlight the range of functions and forms of the letter in antiquity.
As one of the description above mentions, some of these letters are included in the Thesaurus Linguae Graecae, a massive online corpus of Greek texts (subscription based).
Now, in your question, you mention about whether there is "some sort of convention" in letters. For this, the relevant section in the article might be the one titled "Epistolary Theory":
While most discussions of epistolary theory originated well after the development of literary letters themselves, the authors of theoretical treatises or handbooks were legislating for what counted as effective epistolary performance in a literary sense; they prescribed what characteristics should be cultivated in letter formats, and how readers should judge their effectiveness or quality. Malherbe 1988 offers a comprehensive look at original documents in Greek and English, while Poster 2007 gives highlights from a diverse group of ancient epistolary theorists.
Malherbe, A. J. 1988. Ancient epistolary theorists. Atlanta: Scholars.
Including a very useful introduction and summary of epistolary theory, this source book offers English Translations in facing-page format of influential Greek epistolary theorists, including Demetrius (De Elocutione); pseudo-Demetrius (Epistolary Types); and pseudo-Libanius (Epistolary Styles).
Poster, C. 2007. A conversation halved: Epistolary theory in Greco-Roman antiquity. In Letter-writing manuals and instruction from antiquity to the present. Edited by C. Poster and L. C. Mitchell, 21–51. Columbia: Univ. of South Carolina Press.
An essay on letter-writing theory, examples of model letters, and prescriptive guides to writing letters in antiquity.
There is a collection of Plato’s letters, some, if not all, of which are generally considered to be authentic. There are also the letters of Xenophon, again of contested authenticity. But these both do give us insight into epistolographic conventions of the classical period. From Hellenistic times we have a large number of genuine letters on Egyptian papyri, and of course also the epistles of Paul, of which, again, some are generally considered to be “deutero-Pauline”, that is: fake.
From Poster, C. 2007. A conversation halved: Epistolary theory in Greco-Roman antiquity. In Letter-writing manuals and instruction from antiquity to the present. Edited by C. Poster and L. C. Mitchell, 21–51. Columbia: Univ. of South Carolina Press pointed by @luchonacho I have found quite a few rules that I'll sum up here for those interested: