There are 29 unique driving missions in total, so unlike all of the Contracts achievements, this achievement can be obtained simply by finishing each mission as they appear. If you're not up for that, grinding the same mission counts so you can simply find one you like and keep driving it until the achievement pops.
You can select these missions from your phone by pushing
and selecting Driving Contract.
The types of missions from hardest to easiest:
Carrier: pick up a car and drive it to a fixed destination, within a time limit and with minimal damage. These are much harder than they sound because the vehicles can sustain a ridiculously small amount of damage before the mission fails. Colliding with a large object like another car, a bridge abutment, a building, etc. is usually instant failure. Trading paint with another car as you drive past takes about 10% of your car's meter, and hitting small objects like stop signs or plastic road markers takes away 20%. These suck, bad.
Escort: you follow a VIP on a leisurely Sunday morning drive while mobs of sedans crash into, and shoot at, their vehicle. These are made hard by the simple fact that the VIP is in no hurry and the enemy sedans spawn infinitely, so if you can get a breather at all then it's not going to last terribly long. Because they spawn in large groups, attacking one always leaves a hole for the other seven to speed past you and smash into your VIP. The whole thing is a battle of attrition and I never got the VIP home with any more than small slivers of their health bars.
Scamble: maintain a distance of at least 60 meters with an enemy vehicle to increment a download counter to 100%. These aren't generally terribly difficult, the biggest problem with these guys is when they burst steam vents on you because there is a ridiculously large radius within which your car will be thrown around like a toy, giving your target a 200 meter lead while you pull your teeth out of the windshield. If you can smash them and get them stuck (or at least remain stationary for a few seconds), a non-moving enemy vehicle gives you a huge boost to your download counter. The best strategy otherwise is extreme tailgating. If you can remain right on his bumper, you'll cruise right through every hazard he makes by being so close to his car.
Overdrive: apparently the vehicle you're in has turned into the bus from Speed and you have to maintain an arbitrary speed over a series of checkpoints. Be aware that this is not simply momentum but forward acceleration, so something like drifting a corner is not considered "fast enough" for the game because your momentum is angular and not linear. There are a couple of tough ones but they are made difficult entirely by the shitty rear-wheel drive muscle car you're forced to use because they handle the same way in every game: the rear end is completely uncontrollable and they spin out when you accelerate.