First of all, I take no responsibility for any fines you get, or breakage to your vehicle :).
The best way to do a burnout depends on your car, and whether it's front- or rear-wheel drive. The hardest part about doing a burnout in an auto is getting the wheels spinning in the first place. Especially with a smaller four-cylinder motor.
If you don't know if your car is front- or rear-wheel drive, ask your Dad! Or your Mum. Or, try making the wheels spin in the dirt and see which ones spin. If you don't know which wheels are the drive wheels of your car, you can't know how it's going to behave when you start abusing it, and you could get yourself into an argument with some part of the scenery. So be careful!
Front Wheel Drive
My friends and I found this method to work with a few cars. First, find a steep hill, and stop on the hill. Let your foot off the brake, and the car should start rolling backwards. This will give it some momentum going backwards. Once you're moving, stomp the accelerator! When you plant your foot the wheels won't be able to grip and should just start spinning. If the car starts to move forwards, yank on the park brake. In most cars the park brake only works on the rear wheels, so the front are free to keep spinning.
If your car is powerful enough and/or your park brake is good enough, you might even be able to do burnouts on a flat bit of road. You can only try!
Rear Wheel Drive
If your car is rear-drive and a four-cylinder, you don't have a lot of hope here really, especially since four-cylinder rear-drive cars are generally all pretty old now. So if this is you, the only way a burnout is going to happen is by putting a small pool of diesel or engine oil on the road, putting your tyre in the middle of it and planting your foot. This should work, but maybe not for long.
Six-cylinder or V8 cars will be alot easier with a rear-drive auto than a four-cylinder automatic would be. Powerful ones with plenty of torque at low revs will just spin the wheels when you plant your foot. The car might also kind of lurch forwards at first, and most of the weight of the car comes off the back wheels. If you let it lurch and then slam the brake while also mashing the accelerator then the weight might come off the back wheels and let them start to spin. Then you just have to use your foot brake to hold the car still. You have to press the brake pretty hard to hold the car still or creep forward slowly, but not too hard or the back brakes will overpower the motor.
If it's not powerful enough to just spin the wheels on the spot, you could try resorting to doughnuts. This is where you find a spacious bit of tarmac, turning in a circle and going fast enough that the tyres are almost losing grip. Then mash the throttle and if all goes well, the back wheels will spin and kick out sideways. This is where it has to start coming naturally cause it's a little bit hard to explain with text what happens next. See thrashcar for some videos! But be careful, if your car has a tendency to understeer instead of oversteer (most do...), it will plough straight ahead instead when you plant the throttle and you won't do a burnout but you could hit whatever is in front of you. Trust me, I know this from experience!
Good luck with your burnout endeavours, be sure to send pics and video and whatever else.