My last drama was Mr Brain - are Japanese dramas allowed here too?. If they are, then my favourite character in it would be, not very originally, Mr Brain himself, aka Takura Kimuya. He is just the type of obnoxious, genius and yet oblivious character I can't help but adore. On top of it, his antics are hilarious.
As for K-dramas, the last one I completed was... Dream High. Favourite character without a doubt: Pil Sook. The reason is uncertain: possibly I loved her because she made me remember a lot of things. And she was also the best acting out of the lot, in my humble opinion.:)