Who gets the kill when the turret 'last hits' the champion slain?

The kill always goes to the champion who damaged them most recently, unless 10 seconds has passed between death and champion damage, then it is an Execute and nobody gets kill credit.

This counts for champions who die by Turret, Fountain, Monsters and Minions.

While eyeofthehawk's answer is true in the general case, there are certain cases which will award heroes a kill if they are able to buff a tower (most relevantly, Janna's E) or capture point (via Garrison).

If a tower thus empowered scores a kill, it will go to the empowering champion, rather than to an execute, even if the player has not damaged the opposing champion themselves.