In Demolition Derby, you are thrown into a large stadium with several other cars. The idea is to ram anyone you can, as you get a cash bonus every time you damage a car. You must get a specified amount of cash before you or all of the other cars explode. The trick to winning this is to use nitrous to get an immense speed and ram the other cars. To activate nitrous, click .
As you progress through levels, you are given "upgrade points" which you can use to upgrade your car's offense, defense and speed. Focus on the speed upgrades first, then offense, and finally defense.
There is one Demolition Derby activity with 6 levels. I recommend doing this in co-op so you can work towards the "Confidence Men" achievement as well as "Crime Lord".