What is the goal of the League in "The Red-Headed League"?
The fictional Red-Headed League, as Mr Wilson tells Sherlock Holmes and Dr Watson, was founded by a man called Ezekiah Hopkins. He himself was red-headed, and as a result of the prejudice and discrimination that he faced during his lifetime, ordered that his considerable wealth should be used to support men of a similar nature to himself in order to make their lives somewhat easier than Ezekiah Hopkins himself had experienced. Note what Mr Wilson tells Holmes and Watson about Ezekiah Hopkins:
He was himself red-headed, and he had a great sympathy for all red-headed men; so when he died it was found that he had left his enormous fortune in the hands of trustees, with instructions to apply the interest to the providing of easy berths to men whose hair is of that color.
The goal of the Red-Headed League was therefore to use the wealth left in trust by Mr Hopkins in order to help those gentleman who had red hair like its founder, providing them with jobs involving minimal work and effort and recompensing them with a very generous salary. Of course, as the story goes on to show, the Red-Headed League is a complete fiction created in order to get Mr Wilson away from his shop to enable theft and criminal activity to occur.