I assume you're either working on homework, or writing (presumably) YA.
If it's the former, I can't come up with a good reason, and I'm a college bio major so best o' luck to ye. Spiders, the paradigm comparison, still need to rely on either a web to catch the victim or strength to overpower it, because the thing doesn't die quickly enough. With that in mind, think of this from a hypothetical cat's perspective: you're already a very efficient hunter of small prey - if you catch it, it's as good as dead, venom or no - but a big animal like a deer or hippo can still quite easily kill you even if the venom acts in 15s, and in reality it would probably take days in that small a dose. Not much help in either case.
As for writing, I already explained why venom makes no sense for offense.
As a defensive mechanism though, something that takes a little longer would make sense, especially if other large and aggressive predators existed. However, venom is still quite a stretch, because I would think that the venom-producing glands would have to evolve first, then the delivery method, and the former happening in a cat's legs sounds like a disadvantage for the cat - read: a self-eliminating trait in natural selection.
Rather than venom, change it to bacterial endospore that resides on the cat's claws. These bacteria would have a mutually beneficial symbiotic relationship with the cat: they gain a carrier that transmits them to other hosts, while the cat gains a potent weapon. The endospores are triggered by exposure to other animals but not the cat's own system. That makes quick death (<24hr) a bit farfetched - bacterial infection from spores usually takes longer to hit full swing - but by the virtually unlimited viability period of endospores would explain why the bacteria live on all of this species of cat, all the time for their entire lives, without dying off from lack of a host.
Whatever you decide to use, before you go ahead, ask yourself the following two questions:
Why would this come to exist? (you already are so good there)
Why would the cat not kill itself when it licks its claws or scratches off a scab?