It seems to me as if that trigger that you posted should keep adding custom score indefinitely from the first kill that the player gets, because the condition "kills one any unit" will always be true after the first kill... unless the condition is "kills
exactly one unit", in which case it will add score continuously until your second kill, then stop...
If you simply want to show kills in the leaderboard, then simply display Leaderboard (Kills).
If for some reason you want to record the number of kills a player has in custom score, then you want to look at a "
Kills to Cash" system, giving custom score instead of minerals.
There should be no problem with using custom score.
Note that the condition "Always" won't do anything if you have another condition as well, so you can leave it out.
Also note that (your triggers don't make it very clear what you're using) if the trigger is for multiple players, you probably want to use "Current player" in both the conditions and actions of your trigger.
Otherwise keep giving some more detail about your problem.