I am but an amateur, but I hope I can help. While I can't speak for all ancient historians, I can certainly weigh in on one in particular - Herodotus, whose Histories are one of the oldest surviving "inquiries" (to use Herodotus' own description) into the past, what we would today call a "history", dating to c. 440 BC.

Herodotus' Histories were an attempt by Herodotus to find an origin for the Greco-Persian Wars, and cover a large time period of c. 1200 BC, with the rapes of Io, Europa and Medea and the start of the Trojan Wars, to 478 BC, with the Athenian siege of Sestos.

Herodotus used a fairly unique (for the time) method of inquiry, treating history as a full investigation. He supposedly travelled across the Mediterranean, speaking to priests, oracles, librarians and locals to hear their versions of the myths, legends and histories of the various ancient Mediterranean societies. He also travelled across a few nations, Egypt in particular, and made lots of geographical and ethnographical observations while there (such as his observation in Book 2 that the people of Colchis must have descended from Egyptians based on their skin colour). He used all of these methods together, and in his book supplied the differing versions of the same tale before offering his own insight into the issue.

Of course, he was often wrong (hello, reason for the Nile's flooding!), but his method is still pretty solid. He listened to the stories, he read inscriptions and texts, and drew his own conclusions from the data he used. This method is pretty similar to our own methodology today, no? It's notable too that a similar method was used by later historians, both critical and supportive of Herodotus, such as Thucydides and Livy.

In short, ancient historians (at least from the 5th Century BC onwards) used a similar methodology as we have today for studying history; they listened to others' interpretations, they read the sources available to them, and drew their own conclusions.

Sources Herodotus, The Histories, trans G. Rawlinson D. Lateiner, The Historical Method of Herodotus, University of Toronto Press 1989