How did the first and second Hokage died

The First Hokage, also known as the Shodaime- Hashirama Senju died in battle like any other noble ninja would do for their country. (kakazu killed him, even he said they battled in the first ninja war, I think kakazu also took his heart,)

He was revived later by Orochimaru during the Chuunin Exams using the Impure World Resurrection to fight the Third Hokage in the Konoha Arc fuly aware of the situation, but CockOrochi employed a seal on him to rob Hashirama of his free will and so him and Sarutobi fought.

His soul was then sealed by Sarutobi along with his brother -who was in the same predicament- and Orochimaru's arms in the stomach of the Death God or Shinigami

But before that, he and his brother apologized to Sarutobi for all the trouble they had caused.

The Second Hokage, Nidaime Hokage Tobirama Senju, died loyally to his village in fighting ninja wars same as his brother.

Hope that helps 'cause I am NEVER doing research again!
the first i think died in battle between uchiha madara and him ,

(edit) The 1st Hogake fought madara uchiha and won, he returned to the village and died in the war along with his brother.