How is it determined that an ancient language had pitch or stress or tone accent?

In the case of Ancient Greek, it’s actually quite straightforward:

• We know that words had accents, because the ancients made up signs for accents. Words having accents is the norm in language anyway.
• We know that normally only one syllable per word had an accent, because that’s how the ancients wrote their accents. At the very beginning of the invention of accent diacritics, a grave would be used for every unaccented syllable; that eventually went away, and it indicates that the grave was a neutral (unaccented) accent marker.
• The fact that a word normally had only one accent indicates this was an accentual, not a tonal system.
• We know that pitch was involved, because accent systems in the world’s languages are split between stress-based and pitch-based systems. Stress-based systems are normally only stressed/unstressed; pitch-based systems allow for pitch contours and multiple levels of pitch, both of which are represented in Ancient Greek with its acute/circumflex/grave.
• Vedic also had clearly high and low pitch in its accentuation, with regular correspondences with Greek accent, which makes it likelier that the related Greek had the same accentual system.
• For the evidence from Greek writers themselves, I’m going to quote from Allen, W. Sidney. Vox Graeca. the Pronunciation of Classical Greek, Chapter 6:
• Plato in Cratylus 399a differentiates between ‘sharp’ and ‘heavy’ accent (oxys, barys); if it was a stress-based system, barys would have to mean ‘quiet’. which it does not. Plato in Phaedrus 268d uses the same terms to refer to high and low pitch in music.
• The Greek word for accent, tonos, means ‘tension’, and is likely related to the tension of a string in an instrument: the more taut the string, the higher its note. The word is of course our tone.
• The early names for the circumflex were ditonos, oxybarys, symplektos, perispōmenos, “two-toned, high–low, complex, and broken-either-side = circumflex”.
• Dionysius Thrax says that “of course not every word is spoken with the same pitch-pattern (tasis, another word for ‘tension’), but one on the high (oxys) tone, another on the low (barys), and another on both. Of those which have both, some have the low combined with the high in one syllable, and these we call circumflex; whereas others have each of them on different syllables and mantaining their own quality.”

We know that stress accentuation was in use by the late 4th century AD, since Gregory Nazianzen uses stress-based metres in his poetry, and there are hints of stress-based metre in the early 3rd century AD hymns of Clement of Alexandria. Outside of metre, Allen gives no other evidence, and short of contemporary phonetic description, we would be unlikely to get any.