He is largely defined by his persistence, good humor, wit, and his kindness.
Zagreus is often shown to be humble, good-humored, and often sarcastic and self-deprecating. His jovial, if somewhat-insubordinate nature allows him to easily befriend the many people he encounters, treating people of lower status as his equals, although he is respectful of his elders (aside from his father), especially his fellow gods. He is sometimes shown to be somewhat blasé, which has led him to offend the people around him or act without thinking of the consequences. However, he often reflects on this and is quick to apologize when he oversteps. He has a mischievous side, and is fond of pranking others.
Hades is stern, serious, and dedicated to his work. He is shown, in the Remembrances that play before some escape attempts, to have been strict and often cruel with Zagreus during his childhood.
As the embodiment of death, Thanatos has many duties requiring him to venture up into the mortal realm. His most recent such venture took longer than usual - speculated by Hypnos and Zagreus to be related to the ongoing war - but returns some time after the game's opening. He apparently dislikes the mortal realm, in particular how bright it can get.
Thanatos occasionally appears at the beginning of regular encounters and challenges Zagreus to see which of them can slay more shades. He is immune to damage during the contest, and will not retaliate if struck (he'll just complain). If Zagreus either kills more enemies than Thanatos does or ties his total, he will award Zagreus a Centaur Heart in addition to the regular room-clear reward.
After a random number of encounters with the Furies, Megaera will approach Zagreus in his room, and tell him that she's only stopping him from escaping because she has to. (This requires her to have first been in his room to collect her belongings.)