The pokedex sais that Sliggoo evolves during rain.So, if I use Rain Dance in battle,and Sliggoo levels up it will evolve, right ? Does it have to be rain from the overworld weather in order for this to happen ?
Sliggoo has to evolve during rain in the overworld. This means that it does not evolve when you use Rain Dance / Drizzle during a battle.
I heard that it rains on Route 18 (during afternoon), Route 8, and Route 14. So check those routes if you want to evolve your Sliggoo.
Source: Bulbapedia Link, Forum
It needs to be overworld rain; using Rain Dance or Drizzle does not work to evolve Sliggoo.
Source: Experience, and I think other questions like this have been asked.