You do not need a permit to backpack in Angeles National Forest, nor do you need a bear canister. Their are bears there but they tend not to be problematic. You shouldn't leave food unattended, but beyond that it is up to you to decide what level of food protection is needed.
You need an Adventure Pass to park in parking lots within the forest (like at trailheads). They cost $30 annually and can be purchased at Ranger Stations (check hours before you go.. they are not open very often anymore) or most any outdoor gear shop. You need a California Campfire Permit to have a fire. They are free and easy to get online: http://www.preventwildfireca.org/Campfire-Permit/
You can call the Ranger Station for lots of questions. As I recall most of the fire levels within the forest limit campfires to campgrounds. If you want a campfire there are actually quite a few campgrounds in the forest that you hike to. They may also have an outhouse. All of the campgrounds on on this list that are called a "trail camp" are not accessible by car: http://www.fs.usda.gov/activity/angeles/recreation/camping-cabins/?recid=41672&actid=29
An "out and back" type trip is usually good for a first backpacking trip, because it gives you better bailing out opportunities. Finding a trail camp the distance you are looking for from the road would be one good suggestion. You can of course day hike from there. This map: http://www.nationalforestmapstore.com/product-p/ca-1.htm
is good for getting an overview of where these trails and camps are and how much hiking you can do on them.
Water is usually the biggest issue with backpacking around Los Angeles. Many of the trail camps have some form of water (near a stream) but often only seasonally. Outside of ANF; Padres National Forest and Cleveland National Forest are also nearby and offer backpacking opportunities. Catalina Island is also another great backpacking opportunity.
Mt. San Jacinto State Park is another good area to think about backpacking. They require a wilderness permit. The best part is you can take the Palm Springs Tram to the top of the mountain and backpack around in the forest.