As the name of the piece suggests, it was composed and written by an individual named Seikilos.
The arrangement to Seikilos' melody played with a Nyckelharpa, lyre and Choroi flute is composed, sung and played by Marianne Lihannah

Seikilos epitaph is the oldest surviving complete musical composition, including musical notation, from anywhere in the world. The epitaph has been variously dated, but seems to be either from the 1st or the 2nd century CE. The song, the melody of which is recorded, alongside its lyrics, in the ancient Greek musical notation, was found engraved on a tombstone (a stele) from the Hellenistic town of Tralles near present-day Aydın, Turkey, not far from Ephesus. A town that was part of Ancient Greece but is now part of Turkey.
While older music with notation exists (for example the Hurrian songs), all of it is in fragments; the Seikilos epitaph is unique in that it is a complete, though short, composition.
This gives us a glimpse as to what the music of Ancient Greece was truly like and the piece is still being performed today. As the name suggests, the piece itself is an epitaph, probably written to honor the composer's deceased wife or the muse of music.

The Stele is now kept in a museum in Denmark, where I come from, so I thought it fun to translate the beautiful words into a poetic Danish translation as well. Each verse is sung in a slightly different way. As there is a debate as to where the emphasis is in the meter, each verse has a different emphasis because of the different translations in 2 languages.
A possible alternative way of rhythmizing the Seikilos song, in order to preserve the iambic ('rising', di-dum) feel of the rhythm, was suggested by Armand D'Angour, with the barlines displaced one quaver to the right. When I sing the song a second time in Danish, I am using this alternative way.
The verse in English, which I sing first, I sing using the more well-known way of putting the emphasis.
It is a Hellenistic Ionic song in either the Phrygian octave species or Iastian tonos.

In addition to the lyrics, another piece of the puzzle is the inscription engraved on the stele that reads, “I am a tombstone, an image. Seikilos placed me here as an everlasting sign of deathless remembrance

it, which reads in Greek:
eikṑn ḗ líthos eimí. títhēsí me Seikílos éntha mnḗmēs athanátou sêma polykhrónion.
A free translation of this reads: "I am a tombstone, an image. Seikilos placed me here as a long-lasting sign of deathless remembrance."

Dedication
The last two surviving words on the tombstone itself are (with the bracketed characters denoting a partial possible reconstruction of the lacuna or of a possible name abbreviation)
Σεικίλος Εὐτέρ[πῃ]
Seikílos Eutér[pēi]

meaning "Seikilos to Euterpe"; hence, according to this reconstruction, the tombstone and the epigrams thereon were possibly dedicated by Seikilos to Euterpe, who was possibly his wife. (Euterpe is also the name of the Muse of music). Another possible partial reconstruction could be
Σεικίλος Εὐτέρ[που]
Seikílos Eutér[pou]

meaning "Seikilos of Euterpes", i.e. "Seikilos, son of Euterpes"