I think 'I do like you' has more emphasis than just 'I like you'. I think this could be used, in something like, I don't know it's correct, but an example: "You don't like me, but I do like you." that is some like "Despite you don't like me, I like you".

If you just want to tell someone you like them, "I like you" is the most common and natural way to say it. Simple. But, you can say "I do like you" to answer specific questions or to provide emphasis. So it's not really a replacement for "I like you". You might use it to give some background before explaining more. I do like you, but... let's just be friends. You can also use it for emphasis or to contradict what someone else says. A: You don't like me. B I DO like you. (emphasis on the word "do")

In my side ,"I like you" and "I do like you" are the same meaning.It is the degree that counts.The latter is emphasizing but the first one is not.Also,the second one means "I like you very much" or "I like you surely."

Imagine someone asking you 'Do you like me?' you may answer 'I do like you indeed', or 'I do like you as a friend' etc ... 'do' is an auxiliary verb that intensifies or stresses the main verb - do like, do read, don't eat, don't dance ...