Scoreboards can track the kills of specific types of entities.
/scoreboard objectives add < name > stat.killEntity.* < display name >
This increases when you kill a mob of type *. The list of entity types that can go in * are:
However, this method doesn't work if there are two enemies of the same type (2 zombies, 2 endermen). Note that spider is different than cave spider, zombie is different than zombie pigmen, and that skeleton is the same as wither skeleton.
To do the "point-giving" part:
Make an objective called
points (or whatever you want)
/scoreboard objectives add points dummy
Then, set it so that players can see it:
/scoreboard objectives setdisplay * points
Possible values of
list: Displays next to the player name when holding [tab]. (doesn't work in singleplayer)
sidebar: Displays on the side of the screen.
belowName: Displays under the player name (floating above head, doesn't work in singleplayer)
These all work when opened to LAN and more than one player is on.