in order to get spices, you have to kill hell rats which means you have to do the recipe for disaster part of the quest with evil dave... your cat grows up without needing to catch rats
Make sure it is a cat, not kitten when you try to catch the rats (or else you will be wasting ALOT ALOT ALOT ALOT ALOT of time).
It takes about 3 hours for the kitten to grow into a cat. You need to feed it approximately 10 times (you will be given a dialogue box by jagex where it says that you should feed the kitten). When the kitten reaches 100% growth it will turn to a cat. It is recommended to use a ball of wool on the kitten every now and then while you wait for the kitten to grow.
It is also recommended to train a repititive skill like fishing, cooking, fletching (this is what I did), mining or woodcutting while waiting for the kitten to grow.