Like Peachey said, it doesn't make sense to use "like" because cats and dogs don't rain. The origin of the idiom probably has to do with the classic heavy relationship that they have. You can say: it's pouring, it's pouring rain, it's pouring heavily, it's raining buckets, it's really coming down, it's a heavy downpour, there's a torrential downpour...

"It's raining cats and dogs." This is only a metaphor; we never add "like" and use it as a simile.