Do you know any artists that make ancient Greek or Roman music, and how can I make these types of music myself?

There is no writren record nor recordings (of course) of such ancient music. Yes, there are some hints like greek musical modes but they are not trustable in terms of historicity. You could guess greek folk music has a relationship with this ancient music and speculate a little based on archaelogical evidence of instruments, but: nobody knows how roman or greek ancient music sounded.