The first action of my turn is to shoot, which deals just enough damage to make a kill shot, meaning the player then tips his figure over. Can my second action be to shoot the tipped over figure again, thereby gaining an overkill?
Yes page 4:
The 11th point of damage is the killshot. A board with 11 damage will be scored at the end of the turn. (Which means you might have time
to give it a 12th!)"
At the end of your turn, score all boards that have taken killshots:"
this is also how U make a double kill.