How is the prime minister of Russia chosen?

He is appointed by the president (after approval by the parliament).

"The general idea is that the president defines the general aspects of the policy, while the PM (correctly called "chairman of the government") works out the details.
The PM is the head of the executive branch and coordinates the actions of other ministers.

Overall, the PM is considered a rather technical role, not a decision maker. This was not the case though during Medvedev's presidency. While nominally second to Medvedev, Putin (who could not run for a third term as president in a row) was de facto the man in charge."

Same way anybody gets a job in Russia straight up vodka drinking contest. Last man standing.

The lying, thieving, murdering despot Putin organises it and the Russian parliament rubber stamps his choice. Putin selects people who are compliant in swapping jobs with him so that he can get around the law which allows a President two only consecutive terms.

Ouija board.