Quote from Vitamin Z
Use a slime finding tool, such as this one
Slimes are rare and are only found in certain chunks deep below the surface near bedrock layer.
Quote from DevinXIII
Dig down near bedrock and then just make a HUGE room. They should spawn pretty fast.
Also make sure the light level is ok.
Quote from Lilboda
the light level is really dark, i cant get anything to light up down there