Best time to visit Zhangjiajie and where to stay that is nearest to park?

Well, generally speaking, April to October is regarded as the peak season to tour around. So it's ok for you to go there in June or July. I personally prefer June. The temperature won't be too hot.
What's the budget of your accommodation?
You can find a hotel in Wulingyuan Area. Xibujie Inn is a good choice. The price is about CNY 300 per night. It has nice environment and warm service. You can go and see. It's located at 53 Xibu Street. Hope you like.