Vigilante Missions are required for 100% completion of the game. You need to complete at least 20 of the current crimes. They do not have to be done consecutively.
To start the mission, you'll need to be inside a police vehicle. Pull over to the side of the road and completely stop the car. A message will appear telling you to what to push for access to the police computer, so access it and go down to "View Current Crimes" on the menu.
From there, you'll see three random crimes which have been reported. Select the one you wish to attempt and your GPS will then be updated with details.
The mission possibilities are fairly simple. You'll either have a single person who needs to be taken down, or a gang who you need to kill. Occasionally you'll have the option to go after a stolen car, in which case you'll have to chase after it and drive-by the driver and usually the gun wielding passenger too. The single people are the easiest and quickest to complete, but doing it over and over can get boring, so I suggest mixing it up a bit.
When you start the mission, you'll receive a time limit to get to the crime scene. This is usually a few seconds more than you actually need, but you'll still need to follow the GPS and get there as quick as you can. If you don't get there in time, don't worry about it. You can go back into the computer and start a new mission. Once you reach the crime scene, you have as much time as you need to eliminate the problem.
You're allowed to get out of your car to kill the enemies, but not until you actually reach the crime scene. You must arrive in a police vehicle. Remember to pick up their cash once you've killed them. You get no other reward for these missions apart from the cash and weapons you can pick up from dead bodies. Generally, gangs won't drop any cash, but you'll always have a few weapons to pick up should you need them. Single enemies on foot always drop a wad of cash.
Rather annoyingly, the police will still come after you for shooting people, even though you're effectively working for them, however once you kill all of the gang members, you'll instantly lose your wanted level.
Once you've finished a mission, get back in the police vehicle and use the computer to select a new one.
If you don't know how many missions you've got left to do, pause the game, go into the Stats menu and go down to Score, then scroll to the bottom and you'll see a stat for "Vigilante levels done". This needs to be on 20 or more for you to be awarded with the percentage for beating this objective.