Rubbing alcohol (at least 90%) from the drugstore is usually your best choice. It's one of the few substances that will mix with both oil and water. I'd drain the engine till it looked clean, then run your "cheap" oil for a time before replacing it and inspecting it. If there's any sign of contamination, repeat with another type of oil until you find some that are not affected. Then replace the original oil.
If you use regular oil, fill up the reservoir again with new oil and give it a good shake to ensure all of the old oil is out. Leave the car for at least 24 hours before driving it again, to allow any remaining alcohol to evaporate.
The alcohol itself won't cause much harm even if it does escape through your exhaust, but it can contaminate your engine oil which could lead to serious damage down the road. The best option is to simply pour some rubbing alcohol into your tank every time you fill up on gas so you don't have to worry about it. It won't hurt anything and it's cheap insurance against having to change your oil prematurely.
Spray Simple Green or any citrus-based cleaning all over the engine (particularly in the valley below the cylinders) and allow it to rest for a half-hour or more to dissolve the oil before rinsing with water. The heat from the engine will help evaporate the cleaner faster.
If there is still some residue after cleaning, try using Auto Detailing Products such as WD-40 or Turtle Wax. These products are designed to remove grease and other contaminants from your car's exterior surfaces.
If you prefer a natural remedy, use olive oil. It works well for most engines and doesn't require electricity to work. Just be sure to use extra-virgin olive oil instead of regular vegetable oil.
Finally, if you want to go the electric route, here are a few products that will help remove engine oil: Engine Offset Oil Cleaner by Prime Time, Off-Road Oil Cleaner by Advance Auto Parts, and Off-Road Motorcycle Lubricant by Mobil 1.
You may use the same procedure if your oil container is made of plastic. Dove dish soap is reputed to be especially effective. If it doesn't work, measure out 1/2 cup of household ammonia and pour it into the empty plastic container. Shake it lightly and let it for 24 hours before washing. The acid in the soap or ammonia will break down the oil so it can be washed away.
If you don't want to use soap or ammonia, there are a few things you can try. Heat the oil container in the oven for 10 minutes at 400 degrees F. This will evaporate some of the oil but won't break down any harmful chemicals. You could also place it over a burner on low heat for about 20 minutes but make sure not to burn yourself.
Finally, you can buy commercial oil-drainer bottles that are specifically designed to soak up oil spills in parking lots and at sea. These usually have an attached strainer basket and come with instructions on how to use them.
The best thing to do is avoid getting oil on your clothes in the first place. If you must wear oil-stained clothes, wash them immediately in hot water with detergent.
Sprinkle a liberal amount of baking soda on top of the oil stain and softly moisten it with water. Then, using a circular motion, brush the stain until it begins to come up. Once finished, rinse it with water and repeat if required. Baking soda is a natural degreaser that will help rid your driveway of oil stains.
If you want to use soap instead, use a gentle dishwashing liquid rather than one for cleaning cars. It will work just as well and won't cause any problems for your asphalt surface. Simply mix 1 part dishwashing liquid with 2 parts water in a bowl and soak a piece of cloth in it. Gently scrub the oil stain away until it's gone.
Finally, if you want to use alcohol instead, be careful not to get any on anything else that it might affect. Both gasoline and turpentine are extremely toxic if not used properly, so keep this in mind when choosing an ingredient.
Now that you know how to remove old oil from asphalt, go for it! You can reuse the baking soda for future stains if necessary.