I don't think that "kill the clock" means the same as "kill time." When we talk about killing the clock, we are speaking of a very goal-oriented activity. You kill the clock because you want to get to a certain time (the end of the game). Killing time simply means doing idle things because you don't really have anything you need to be doing. It's much less focused or goal-oriented than killing the clock.
I too think of "kill the clock" in sports terms. Usually the leading team will try to run time down so they can keep the upper hand. You might look at the same as the idiom "just killing time." In a classroom setting, a student might be trying to "kill the clock" by wasting time until the bell rings. The student might try to distract the teacher or turn the discussion subject away from the topic at hand. I had several students who would do this in an attempt to stretch out a fun activity or to postpone a less pleasant assignment they knew was coming up. They also had the mistaken impression that killing the clock would mean I didn't have time to assign homework.
In football and basketball, to "kill the clock" means to stop the game clock. Both sports use a clock for time limits (for each quarter or half). As time is running out before the end of a period, teams often try to get in an extra play or try to score before the end. Teams try to "kill" or "stop" the clock by calling a timeout; or, in football, by running out of bounds or deliberately "grounding" the ball (to stop the game clock). Using the clock wisely in this manner gives the team an extra play or two in order to score last second points.