When playing Kennen, if I play my Chosen Champion, will it become Empowered?

Yes — playing your Chosen Champion from the Champion Zone triggers Heart of the Tempest's ability and Empowers the Legend (Kennen's Champion Legend).

"Empower me" refers to the Legend itself, not the unit you played — so your Chosen Champion doesn't gain the Empowered status from this; only Heart of the Tempest does. By default you can play cards from your hand or your Chosen Champion zone (Rule 419.1.a), and the Champion Zone is not your hand, so playing a card from there meets the trigger condition. If you play a second copy of your Chosen Champion from your hand instead, that's playing from your hand, so the Legend would not Empower (FAQ #11861).