I will leave from about the 15th of June to the 5th of July 聼起來自然嗎?

'leave' is something we do at a single, point, time. If you want to say that you'll be away for a period of time, then you need to use a different verb.

I will be away from about the 15th of June to the 5th of July.

If you don't know exactly when you will be leaving, then this might work:

I will be leaving sometime between the 15th of June and the 5th of July.

I suspect you mean the first one.