Difficult answer as it depends on the situation...
The most difficult part is getting the person some place where they wouldn't have told anyone. Options include following them when out one night and picking them up in a fake cab when they are drunk, pretending to be a woman (internet-style) and meeting them some place or simply kidnapping them at some point.
Of course, you'll need somewhere in the middle of no-where to perform the kill. Some place out of ear shot. Good places include on a boat at sea or in the middle of the countryside.You'll need to check out the area and learn it like the back of your hand if you do it in the countryside. Make sure no-one sees you, ever. If they do, find some place else.
Weapons: well, a gun requires a license and bullets are kinda tracable, knives and blunt objects are messy and leave DNA everywhere. I'd go for a compound crossbow. These are quiet, you need no license to get one and the projectile can be recovered afterwards. Rate of fire is poor, however. Give it 1 year + after buying the weapon before you use it.
Disposal: dumping at sea covered in chains or burying somewhere. Burying is a pretty good option if you go some place very remote. You can get DNA in the tranport vehicle if the victim is already dead, however. Burning the body will remove any DNA from yourself. Make a a decent fire.
Another method would be to simply shoot the person at their house. Drive up to the door (stolen car or plates copied from an identical car to yours), detach the doorbell ringer and attach to long wires to the switch. Run wires to the car. Aim up your weapon of choice (crossbow, prolly) and close the connection with the cires. When they answer, fire and drive away.
Just some ideas.....
Of course, I think killing is a truly terrible thing, although if someone did something horrific to a member of my family, I'd probably have to do soemthing about it.