"Life finishes" vs "life is finished"

In his life finishes, finish is used as an intransitive verb. In his life is finished, finished is used as an adjective. Both are correct. Neither means he is the performer of the action.

In I am done, done is used as an adjective.

You've expressed a wish to use the word "finished", but it's not the word a native speaker would use.

More common phrases are:

His life is over.

His life has ended.

If you want to convey that the person is responsible for the action of ending their life, you could use:

He has ended his life.

Be careful - that means suicide.

My suggestion would be to use "end" E.g:

His life ended