After completing the mission, "Truck Hustle" you unlock the option to complete assassination missions. You will notice a grey cross hair on your map. This is the location of the pay phone that you will get your mission from. Go to the phone and answer the call, and you will be given instructions on which target you will be taking out. Once at the location, exterminate the target by any means you feel most comfortable with. All that matters is that they die.
The assassination missions you receive will be in a completely random order, so the numbering below is simply for organizational purposes. After receiving your instructions, the contact will provide you weapons and armor. Hail a taxi and have them take you to your target.
1. Migration Control: From where your cab drops you off, run up the sidewalk east towards the helicopter. Once you get about halfway there you should be able to see it past the trees and such. Pull out your rocket launcher, and send 3 rounds at the helicopter to blow it up.
2. Bailing out for Good: Your target will be walking out of the police station towards a waiting car. Be fast about shooting him because he will quickly get to his car, and then you'll have a high speed chase to deal with. When you shoot him, a couple of his friends will shoot back at you, so stay alert.
3. Derelict Target: You will have 3 targets in an abandoned building. Instead of using a taxi to get to your destination, use a helicopter. Land in the street where your taxi would usually drop you off. As you look down the alley towards the building, on the 3rd floor you can see one of your targets looking out the window. Sniper him off. Next go after the target on the ground floor. He will be standing by a group of cars. He will be alarmed by you shooting his friend, but if you're quick enough, shoot a rocket at the cars to blow him up. If he escapes, he will go to the basement of the building, so just walk into the ground floor, then toss a grenade down the flight of stairs toward him. Then go back to your helicopter. Your final target will be on the top floor below the roof. Land on the roof, and look down one of the holes to shoot your final target. Then just take off.
4. R.U.B. Down: There is a group of bikers in the Acter Industrial Park having a drug deal. When you cab drops you off across the street, pull out the rocket launcher and aim for the middle of them. If you try to kill them individually, they will all get on their bikes and scatter, making it a much more difficult mission.
5. Taken Out: You will see a stretch limo that needs to be eliminated. Send a rocket at it to complete the assassination.
6. Industrial Action: When you arrive, take a look at your map. You will see a group of targets all together, then one off by himself. Go to the lone target, which will take you around back of the buildings and up a flight of stairs. After dealing with him, you will see in front of you two large structures connected by bridges. Pull out your sniper rifle and take down as many targets on the bridges and buildings as you can see. After you can't see any more, work your way up the building in front of you eliminating the enemies there, and by aiming across to the other building.
7. Hook, Line, and Sinker: Your target will be in a boat off the northern coast of Alderney. Follow the GPS path to the target and you will find a boat waiting for you. There is an invisible barrier protecting it from any rocket fire until it is engaged. The best way is just to gun it down with your SMG while chasing it.
8. Dead End: Your target will be driving a silver Patriot. One well aimed shot from your rocket launcher is all it will take, as your taxi will drop you off across the street, giving you a perfect view of your target.
9. Water Hazard: Your target will be just offshore in a boat. From the taxi drop off location, walk towards the water. You will see the target right in front of you. You can either blow up the boat with a rocket, or just sniper off the target if you'd like to be more subtle. You don't need to worry about wanted level, if you get any, it disappears when the target is dead.