What is Lin Zexu's goal in writing the letter to Britain's Queen Victoria

To appeal to her integrity and ask her to stop harming the Chinese people

His goal was to appeal to Queen Victoria in order to slow down (ideally to cease) the opium trade in China and to un-yoke themselves from Great Britain in that economic regard. For his actions, he was both lauded (initially) and scolded (after the fact) by the Emperor, whose own power was limited in dealing with the issue at hand.