Mifune is a prideful leader who is prone to fits of anger, especially when it concerns Tylor. Whenever he becomes angry, which is often, he takes out his sword and will slash at something if he is not stopped. Throughout most of the series he deeply resents Tylor and all of his accomplishments, so much to the point that he nearly attacks Tylor himself before being restrained by Yamamato. He doesn't seem to have a grudge against the rest of the crew, and in fact recognizes that they would have incredible talent if they had a different captain.
In the later episodes, he starts to see Tylor as an opportunity for getting out of various situations. When faced with a sudden war in episode 23, he and Fuji acknowledged that Tylor was the only one with a chance of winning. In the last episode he begrudgingly admits that Tylor is important to the military.
Even though he works with Fuji, half of the time he doesn't listen to Fuji's advice and will cockily think of another way to get rid of Tylor.