How the akatsuki die?

Sasori died from having the only remaining human part of himself stabbed.

Hidan has yet to be confirmed dead, but is slowly wasting away under the Nara grounds.

Kakuzu had all five of his hearts destroyed by Hidan, Naruto and Kakashi.

Deidara turned himself into one of his clay sculptures and detonated himself (in vain) trying to kill Sasuke.

Itachi died from a terminal illness and the strain of having to "fight" Sasuke.

Nagato died after using his final technique to restore the lives of everyone that had died during his invasion of Konoha.

Kisame, after being weakened by a group of Konoha shinobi, broke free and ordered his summon sharks to eat him alive in order to keep the enemy from gaining information about Akatsuki.

Konan died as a result of Madara stabbing her through the abdomen with a metal pipe in order to information on the whereabouts of Nagato and Yahiko's bodies.

Zetsu and Madara are currently alive.

If you were asking about the organization as a whole, it has yet to completely dissolve.