I am still in doing my undergrad in the states- I have a year left, I just wanetd to know ahead of time for Law School what the "ideal". L.L.M candidate looks like on paper. I want to study at either Cambridge or Oxford.

Hi Trojan,
If you want to school at the best, you'll definitely have to be one of the best. In the first place, u have to fairly estimate yourself with your contemporaries in your own country. If you're in the top class, then you're on your way.
Good (detailed) references from ppl/professors who are alumni/alumnae of the school u wish to attend would be v.helpful; and, you have to write a v.good essay.
For U.S schools (and maybe UK schools too), volunteer work is also important. Depending on the competition, some relevant work/research experience might be key.
Hope this helps...
Cheers!