how does pierre become count bezuho

I believe that he inherits all his father's wealth after he dies.

Because Pierre was illegitimate, he would ordinarily not be able to inherit his father's title and wealth.  However, before falling ill, Count Bezukhov had written a letter to the Emperor, requesting hime to legitimize Pierre, in addition to writing a will bequething his belongings, etc., to Pierre. During the count final illness, Prince Vasily knew of both the will and letter and tried to conspire with Pierre's cousins, the Mamontov Princesses, to suppress both documents, but Princess Drubeskaya (knowing that Pierre was more pliable than Vasily and the Mamontov princesses) interceded.  Thus, the documents reached their intended receipients and Pierre inherited title, property, wealth, etc.