What happens to the coins on a killed character?
I've not played or heard of the game before but reading the rules file On BGG here this seems to be covered in the Resolution phase example.  (and probably should be under the 'eliminate a card' rules section as well.

The example given shows that influence on an unrevealed card is discarded when the card is discarded.


  Blue's card is a Soldier which must eliminate and adjacent card. 
  Blue chooses Red's second card, and reveals it.  Blue discards the
  influence on it.


I think this means that no player gets the influence on the discarded card.