Two points: - There is a grammatical rule, widely ignored though it may be, that says never to end a sentence with a preposition. Thus, your sentence, ideally, should not end with "on". - "continue to talk" is fine but I think "carry on a conversation" is better [using "a" rather than "the" as Michael suggested]. To fix these, I suggest "How can I carry on a conversation with a girl upon whom I have a crush?" Or, more poetically, "How can I carry on a conversation with a girl by whom I am smitten?"

What you said makes sense, but to sound more natural you cam say, "How can I continue to talk/communicate with the girl I have a crush on."

You would be perfectly understood with this. I do have two small points to make your English seem even more natural. ‘The’ is called the ‘direct article’ which implies some context where both people know what conversation you are referring to. In which case you probably wouldn’t mention the girl again in this sentence (it is already understood). If it is the first time mentioning the conversation or it was a general question the indefinite ‘a’ is better. Also, you can just say ‘my crush’ and you will be understood to mean the person you have a crush on. Generally the briefer your sentences the more natural you sound.

Your answer sounds fine, Masao. You could also simply say: "How can I continue communicating with the girl I have a crush on?"

I would probably write it this way (but yours sounds ok depending on context): How do I keep the conversation going with the girl I have a crush on? How do I get the girl I like to keep talking to me? How do I continue talking to the girl I have a crush on?