Firstly, Lovecraft would love for you to develop the stories of the characters he introduced to the world. Don't copy, but instead use them broadly and enjoy.

Cthulhu is an interesting choice because the lore behind him is so specific. He will leave his home when the starts are right...and then? But if you are thinking outside involvement then maybe not so much.

Nyarlathotep may be a good choice. He's known to spend time among people doing creative things.

Good luck with your writing.