What makes Urahara specifically brilliant rather than just powerful?

Most powerful characters in Bleach win through overwhelming force or unique ability. Urahara wins through information advantage, long-term preparation, and the ability to construct situations where opponents' choices all lead to outcomes he has already planned for.

He is consistently several steps ahead — not because he is lucky, but because he does his homework. He understands his opponents' abilities, psychology, and likely responses before engaging. His combat is really the execution of a prepared plan, not improvisation.

What is the Hogyoku and why is it important to his character?

The Hogyoku is a device Urahara created — an object capable of dissolving the boundary between Shinigami and Hollow. It is one of the most consequential inventions in the series, and Aizen's quest to obtain it is a primary driver of the early plot.

Urahara's creation of it reveals the scope of his ambition and his willingness to work in morally grey areas. He then hid it inside Rukia's soul without her knowledge — a choice that has real ethical weight. He is protective of people, but also willing to use them as instruments when he believes the stakes are high enough.

How does he train Ichigo?

By nearly killing him. His training methods are ruthlessly effective and deliberately brutal. He does not reassure, comfort, or sugarcoat. He puts Ichigo in situations where learning the required skill or dying are the only outcomes, trusting that Ichigo's will to survive and protect will activate the necessary growth.

This is also revelatory of his character: he believes in people's capacity to exceed limits they believe are fixed — more than they believe it themselves.