As there are several hundred characters covered in this question, it is not possible to simultaneously be accurate and cover the topic completely. The one accurate generalization one could make is that different characters faced different challenges in a variety of ways. Even within a single work by Homer, a single character can devise many different strategies for different situations.
Within the Odyssey, for example, Odysseus is generally characterized by the epithet "polytropos" (clever in many ways). He often uses his intelligence to come up with clever ways to wriggle out of difficult situations, such as being trapped by Polyphemus. On the other hand, some of his solutions are grounded in physical strength and skill, such as the archery contest at the end of the epic, and others in willpower and self-discipline. At times, he relies on the aid of Athena and thus his piety.
The Iliad is the story of a war, and many characters face challenges using bravery and physical prowess to defeat enemies. The elderly Nestor uses his wisdom and experience to help the Greeks. Hector displays moral as well as physical courage and consults with his wife when faced with major decisions. Achilles is physically strong but impulsive. Odysseus is distinguished by being clever and sneaky, using persuasion and even lies to further his ends.
In Hesiod's Works and Days, Perses in motivated by greed and the narrator by a sense of justice. They act through legal maneuvers and devotion to or avoidance of farm work. The gods and heroes in Theogony have various motivations and means of making decisions.
With Sophocles and Euripides, violence is not portrayed directly on stage, so audiences are shown the decision making process leading to resolutions of dilemmas. For characters such as Oedipus, solutions are often found in a quest for knowledge or understanding of the past. One step that occurs in many plays by Sophocles, who was himself a priest and known as pious, is consulting oracles. Often characters in Euripides are strongly motivated by personal emotions.