if akali moves to a BF and targets a unit for 1 damage. akali deadly weapon. and i react and kill akali with a spell. does her 1dmg still go through with her gone?

Yes — the 1 (or 2 if Empowered) damage still goes through.

Akali's "When I move" trigger doesn't depend on her staying on the board. The battlefield she moved to/from is information locked in at trigger time, and the target is chosen at finalization, so killing Akali in response doesn't make the ability mistarget or fizzle — it resolves and deals the damage to the chosen unit (FAQ #11760).

One follow-up worth knowing: if Akali was your only unit there, her death means no Attacker/Defender remains, so no Combat Showdown opens and no Combat Cleanup happens — the marked damage just stays. If a Showdown does open (another unit of yours is still there), the non-lethal damage would be healed in Combat Cleanup.