Does Red Akali deal marked damage when combined with Akali's Champion ability?

Yes — when the Akali legend (Rogue Assassin) moves Akali, Deadly Weapon out of a showdown, her "When I move" trigger fires and deals real damage: 1 (or 2 if she's Empowered) to a unit at the battlefield she moved from. That damage is marked on the unit as damage counters and counts as damage marked by you (Rules 142, 417.1.b).

Two quick caveats:

• The damage amount depends on Akali, Deadly Weapon's Empowered status, not the legend's. Empowered legend just lets the moved unit ready.
• The trigger resolves after the move completes, before any cleanup/heal window — so the damage stays marked and can finish off an already-damaged unit (FAQ #11755).