I've driven to and from LAX numerous times and probably the quickest route that I take is the aforementioned 105->605->91->5. Go to www.mapquest.com for detailed directions. Fortunately for you, you'll be going on a Saturday afternoon, so traffic should be ok. Although there's still a chance of a traffic jam, as it is still L.A. Should probably take about 45 mins.
As for driving through Watts on the freeway, don't worry - it's perfectly safe. I didn't realize this was driving through Watts until the poster mentioned it. You'll be on an elevated freeway and won't even notice the surrounding city anyway, as much of the view is obstructed by walls on both sides. The only thing you might want to avoid is getting off the 105 Freeway for a food/restroom stop. This is actually something we did at night (Vermont Ave), and I was wary of the neighborhood, especially at night. I was glad to get back on the freeway. However, it wasn't that bad, and certainly better than some of the inner cities of the Northeast.
An easier (but a little longer) way with fewer freeway changes is the 405->22 approach. Just be sure to to stay in the rightmost 3 lanes after Long Beach, as the 22/405 split happens at Seal Beach and there and there are 6-7 lanes of traffic in each direction