Hi, I am going to Taiwan on 19 Jun to 25 jun 2016. may I know what is the weather conditions?

It may be cloudy, and rain often at that time. The temperature may range from 80F to 90F. You are advised to bring T-shirts, shorts, dresses, sandles, and an umbrella.
To be honest, it is not an ideal time because the typhoon season would last from June to September every year. The best time lasts from March to May, and from October to December. So, you may change your plan a little bit. ;)