I too am learning han'gul, which I am getting great satisfaction from.
From my study material I was surprised to learn that korean is grammatically related to japanese and mongolian. Korean is thought to belong to the Altaic languages, meaning that it is also related to Tungusic and Turkish.
Grammatically korean is totally different from chinese. There is no connection between them. However, many korean words (as opposed to gammar) come from chinese, in the same way many english words come from latin.