I would start as soon as possible.
Hi, I'm an LA native... born and raised... Give the "Mid-City" area a shot, as well as West Hollywood. There's a lot of great apartments near Melrose Ave. area (Melrose and Fairfax) as well as around 3rd street (3rd and Fairfax, to La Cienega and 3rd). You might even find some good deals in Beverly Hills, believe it or not. And if you check out the homes in the neighborhoods off Beverly Blvd (Bet. Beverly/ La Brea to Beverly/ Wilton) you can find a lot of great deals on guesthomes/ back houses.If you really want to save dinero, you can commute from the "Valley" (North Hollywood, Studio City, etc.) but I leave that up to you. Another option would be to live in Downtown LA... that would be going just east of the 110 freeway, and while this suggestion contradicts my earlier point, I'm only suggesting it in the event that you get an internship or other opportunity with a downtown law firm or courthouse... it would be great to be close by!Also, give http://aptmag.com or http://losangeles.craigslist.org/ a shot to get an idea of apts. in the area.Good luck, and have fun at UCLA!