how are state representatives to the UN elected?

In the US, the UN representative is appointed, but must be confirmed by the Senate.

Unless it is a recess appointment, which is how we got John Bolton as our UN rep, appointed by that other guy, even after the confirmation hearings showed severely eroding support for him, and after Bolton said things such as:

There is no such thing as the United Nations. There is only the international community, which can only be led by the only remaining superpower, which is the United States.

and

"The Secretariat Building in New York [where the UN is housed] has 38 stories. If you lost ten stories today, it wouldn't make a bit of difference."

Edit @Jaker: Oh, please. Obama appointed Rice, and she was confirmed by the Senate.

Bribery

Appointed by the President and confirmed by the Senate unless it is Obama who will ignore that Constitutional requirement.