Legend has it that the Jade Emperor wanted to choose twelve animals which will be represented and will be designated for each year. He decided to have a race to decide the order of the twelve animals whose names will come to the order of twelve.
There are two variations of this tale but all resulted to the same fate for the cat...
Version One
The cat and the mouse were good friends and neighbors that time. When the two animals heard the news they both become excited and the cat told the mouse that they should wake up early in order to be the ones who would be signed in first. The cat asked the mouse to wake him up because he usually wakes up late. The rat promised that he would wake up the cat early and they would go together.
The next morning the mouse was too excited and forgot his promised to the cat and immediately proceeded to the gathering place where he met tiger, the rabbit, dragon, snake, horse, sheep, monkey, the rooster, the dog and the pig which run a lot more faster. In order not to be left behind he thought of a good idea. He approached the straight forward ox and promised him that he would sing him a song if the ox allows him to carry him. The Ox and the rat arrived first but the mouse slid in front of the ox and became the first one to be enlisted on the twelve chose animals. The other animals followed and the twelve animals were complete.
Meanwhile, the cat was too late when it finally arrived, the selections were over. This is the reason why the cat hates the mice so much that every time they meet, the cat will always chase and kill the mouse.
Reference:
Origin of Chinese Zodiac
Version Two
The cat has forgotten the time of the race and asked the rat for the schedule. The rat very alertly gave the cat the wrong time.
Towards the end of the race, a river had to be crossed. The rat asked the ox to ferry him across. When they arrived on the other side, the rat jumped over the ox's head to finish first.
The ox came second, followed by the tiger, the rabbit, dragon, snake, horse, sheep, monkey, the rooster, the dog and the pig. The cat missed the race and swore to kill the rat which, so the legend goes is why cats chase rats.
Source:
How The Rat Jumped Ahead To Win The Race