'I do like you' confirms the fact ,that the person likes the other. 'I really like you' emphasizes this fact. If someone asks the other ,whether they like them or not ,the answer will be : " I do like you". It is usually a reaction to remove doubt or confirm the feeling . " I really like you" will not necessarily be a reaction or an answer to confirm ,rather a statement to emphasize the expression of liking the other person. It is mostly an initiated statement by the speaker to draw special attention to the fact.

I do like you = negates a negative (think about it) I really like you = like you a lot

I do like you..just confirms that yes you like someone. I really like you...really is a degree, it means you don't just like the person, you like them a lot., 你喜欢我吗？， 对/是，我很喜欢你。Do you like me, yes I do like you. 你喜欢我吗？对/是， 我非常喜欢你/我真的喜欢你 Do you like me? Yes, I really like you.

The difference will be explained by not explaining it by asking a question to explain it.