Captain Whitfield's the "Man," but in a totally good way. He's cultured (he reads poetry and plays the violin), but he's also all about working hard with his hands, whether it's manning his ship or farming his land. Think of him as an American Renaissance man.
That's why he's all into giving Manjiro—whom he adopts—a "proper upbringing" (3.15.65), which to him means these things: "A boy should have land to roam, work for his hands to do, a pond to fish, and a horse to ride" (3.15.48). Oh—and a mother, too, which the Captain promptly fulfills by marrying his girlfriend.
The Captain's an all-around perfect man: family-centered, responsible, sensitive, and open to foreigners. If we don't learn much more about him, then it's because he's the consummate good guy in a supporting role. Plus, it's not like Manjiro is necessarily that complicated either and he's the main character.
So think of Whitfield as the model of what a man and captain should be in the book; he's the best that America offers to Manjiro, the perfect embodiment of the country's best values, particularly respect for all people, regardless of class and race:
He was the very most important person on the ship, and he managed the John Howland with authority, and yet he treated everyone with kindness and respect, no matter what the person's rank. (2.10.14)
Like the boy he takes in as his own son, the Captain is good to the core. No wonder they like each other so much.