There are two ways I have seen:
My scholorship letter was in the same envelope with my admissions letter and states that I was awarded it due to my academic achievement and my ability for success in law school. In these instances I don't believe that there is anything that you do- but just how the admissions team feels about your packet.
Secondly, the law school I applied to has two full tuition scholarships that students have to write an essay for. One is solely based on undergraduate acheivement, leadership skills, and Lsat scores. The other is based on undergraduate acheievment and public service. Finalists are chosen from the entries and then someone is chosen to recieve that scholarship.
I don't know if there are other ways to get scholarships from the law school itself- this is just my personal experience.