A license for you is not needed to ride off road.
California is different from most of the other states in that it requires a dirt bike to be licensed to ride on public land. If you ride at a private track no bike license is needed. You should probably read this regarding the California "Green Sticker" (license)
http://www.dmv.ca.gov/vr/ohvredgreen.htm
If you are going to be here in the summer months then you should definately get a "Green Sticker" instead of a "Red" sticker bike.
If you are only going to be here for a few months it will need to be a quick purchase. If your family is willing to help out they can handle selling it when you are done.
The best way to buy a used bike is from the website "Craigslist".
http://losangeles.craigslist.org/
You should be able to access that website now and start looking at the prices of bikes. I highly recommend buying a bike that has the proper "Green Sticker" and preferably a "Spark Arrestor". If you have those two items you won't be limited in where you can ride. You might be able to find a deal where someone is selling all the riding gear along with the bike. You MUST have a helmet and I highly recommend boots, gloves, goggles and a chest protector.
Santa Monica is on the shore just west of Los Angeles. The entire area is like one giant city, however, where dozens of cities have all grown together. I consider the traffic in Santa Monica to be the worst in the state.
You will need to transport your dirt bike to a suitable riding area so you will need either a truck or a vehicle with a trailer hitch and a small trailer. The logistics of getting a suitable vehicle might prove difficult for such a short stay, unless your family already has one that you can use.
If you are going to visit soon (June through August) you will be in our summer period, which means it can be fairly hot. Riding local tracks will be fine or the mountain areas that allow off highway vehicles. If your visit won't be until September through May then the desert would be a great place for you to ride (it is way too hot during the summer). From Santa Monica the desert is a few hours drive but then you will be in an area where you can ride all day and not cross your path. If you ride in such a place, however, be sure that you do NOT ride alone!
As for your question of if Los Angeles is a good place for a dirt bike interest:
Yes, it is. Other than some private tracks there aren't many riding areas close by but there are some great riding areas a few hour drive away. It is much better to live close to where there are good jobs and then drive to where you want to play.
Rod McInnis