I got this achievement on the first mission by Joe's Apartment (Going Out of Business). I used a non-upgraded car, fairly standard speed. You'll need grenades and molotovs, which you can buy at Harry's and Giueseppe's, respectively.
"Just a tip, there are actually molotovs inside the garage where you blow up the cars. They are on the right of the door if you're facing into the garage." - Thanks MEETSP1NNER
Start the mission, follow the red line and at the turn into the yard, do a Drift. Then gun it and follow the line straight - you should get a few Velocity points, then hit the ramp for a Jump. Turn left after landing and get a Vehicular Kill on one of the enemies by your objective, then spin the car and hop out behind it.
They'll be 3 enemies and 3 or more will come out after you destroy the cars. In no particular order, get an Explosive Kill (grenade), Burning Kill (molotov), regular Kill (a firearm kill without a headshot), Headshot, and a Melee Kill (do this one on the last of the first 3 enemies before destroying the cars so no one else is shooting at you).
Go inside and blow up the cars for Demolition, then go outside and get other score actions you didn't get on the first 3 enemies. Then just go hit the safe zone for the achievement.
(Please note that this is just to give a general idea of how to get this achievement - feel free to alter it. For example, if you don't get a Velocity bonus at the beginning, before going to the safe zone grab a fast car and hit a long strip of road)