Many poems in the ancient world were intended to be sung aloud to an audience. Lyric poetry.
Did these poets write their own music, too? Did they perform their own poetry, or were they the modern-day equivalent of songwriters having their poetry/music sang by other people? Does any of this music survive (in some form)?
Roman poets like Horace and Catullus normally recited their poems, rather than singing them. Singing was the norm for Greek poets, but after 300 BCE or so there started to develop a more literary and scholastic form of written lyric poetry, which happens to be the type of lyric poetry that's best preserved: actual songs from after that point are poorly attested for a few centuries. The famous Roman poets were trying to develop on that scholastic tradition, rather than the musical tradition. Their poetry did get set to music every now and then in the mediaeval period, and more recently too, but that's not how they were originally performed and read. The old classics were still performed musically for a long time, though (think of the stories of Nero's musical performances).
For the classics that were performed to music, yes, the poets were musical composers as well. None of their music survives. In some genres, especially epic, it's probably wise to think of something closer to chanting than a melody that the people can hum (this clip of a performer in a modern oral epic tradition is likely to be a good starting point for your imagination, aside from the different rhythmical structure and the different musical instrument). But in lyric, it's likely that we should be thinking of something a bit closer to a modern western-style melody: the strophic structure of choral and epinician poetry, in particular, suggests a melody that was repeated in each stanza. Certain poets, like Timotheos, were known for their musical innovativeness as much as for their lyrics.
We have only slender fragments of the musical theory that started to appear in Athens in the late 5th century BCE, and the only sample of ancient Greek musical notation is the Seikilos epitaph. Even there, though we have indications of musical notes, we have no information at all about performance practice, ornamentation, tempo, tuning, accompaniment, or style (all the reconstructions based on the Seikilos epitaph are extremely speculative and extremely unlikely to resemble an actual ancient performance). There have been suggestions that the tonal nature of the early Greek language was reflected in musical melodies, but that's conjectural.