Han dynasty is one of important dynasty in Chinese history .

Han dynasty is one of the most important dynasties in Chinese history. (or) Han dynasty is an important dynasty in Chinese history.

there are a lot of achievement that opened the door to other civilization for the first time and made great progress in foreign trade .

There are a lot of achievements that opened the door to other civilizations for the first time and made great progress in foreign trade.

In this dynasty ,The silk road already lead to middlewest Asia,even Roma.Arts were also very prosperous and occured lots of prominent works about culture ,history and philosophy .

In this dynasty, the silk road already lead to mid-west/middle west Asia. Even Roma Arts were also very prosperous and (occured probably doesn't work here) lots of prominent works about culture, history, and philosophy.  

In 100 A.D ,the first dictionary was compiled ,which contained about 9000words and provide paraphrase and illustrated different spelling ways.Simultaneously ,

In 100 AD, the first dictionary was compiled which contained about 9000 words and provided paraphrases and illustrated different spellings simultaneously. (I would omit 'simultaneously')

There are also made great progress in technology ,invented pepel,waterwatch,sundials and the machine for predict earthquake .afte 400 years ,the dynast's corruption prompt the country to die.

They also made great progress in technology. They invented the pepel (I don't know what this is), water watches, sundials, and the machine that predicts earthquakes. (this is called a seismograph) After 400 years, the dynasty's corruption prompted the country to die off.