When I play a champion, do they come in exhausted?

Yes, when you play a unit (including a Champion), it enters the battlefield exhausted.

According to the general rules for playing units, all units enter the battlefield exhausted unless a specific card effect states otherwise. This is consistent with the ruling found in the game's FAQs:

• From the FAQ: "Yes, the unit enters exhausted. All units enter exhausted unless otherwise specified."

While some specific units may have text that allows them to enter ready, the default state for any unit entering play is exhausted.