If you look at the HUD when playing the game, you'll notice icons above your EXP bar as Zero. One is your kill skill and should have the same icon as the skill in the skill tree. When it activates, it'll show up for the duration of the kill skill, then disappear when it's done. Kill skills stack with other kill skills, but do NOT stack with themselves, only refresh the timer.
You'll get a similar thing with Zero's Move Like Water skill. When you attack with a melee weapon, you'll see an icon in the middle of the EXP bar show up and start to tick down. This is how long you have to make a ranged attack to get the bonus from Move Like Water.