This seems pretty useful, however, you want to bind a key to two commands... The solution to that was found on stackoverflow.
The answer is to use eval, so in your
.screenrc insert the following (as an example, you can change the binding to your liking):
bind k eval "next" "prev" "kill"
User xhienne also observes that you do not need quotes when using Control-A, as such:
bind k eval next prev kill
You are binding the three commands next, prev and kill to the letter k (so you hit control-a then k). At first I thought I solved the problem with bind k kill next, but after some testing it became apparent that it was in fact doing as the manual says (which is to fall back to the previously displayed window, not the previous or next window)
This answer updated as per xhienne observations (which are correct) below. They note that since it displays the previously displayed you need to go to the next window, then previous, then kill that window in order to get the posters desired behavior.