Meaning of "I'm kind of on a clock here"

Solution 1:

I remember this scene vaguely. Phoebe was basically saying she didn't have much time left since she had already gone into labor. What she wanted was for Rachel (or was it Joey?) to ask Frank if she could keep one of the triplets she was having for him (yes I'm a major Friends geek). The question obviously had to be asked as soon as possible.

So basically being "on a clock" in this context means not having much time left for an action to be taken, or being in a hurry, as you said.

Solution 2:

There are two slightly different meanings; "on the clock" normally means that you are working or that the time is costing you something. By analogy to being clocked-in at work or to a taxi/cab meter running.

But "against the clock" or "up against the clock" is used when you are working against a deadline - as in this case.

From a strictly accurate point I would say the line in the show should have been "I'm up against the clock here"