Haha...now THIS is an interesting topic! I'll assume that you were seriously asking and give you a serious answer!
Well, let's see... you'd have to develop some sort of superhero-like moral code, otherwise you'd become just like the bad guys! Just as the law and politics can get in the way, so can shit like paralyzing some dude who broke into your house, only to later find out his wife needed some kind of expensive surgery and that was the only way he could think of to get some cash.
The probability of that is like ZERO, haha, but my point is that you NEVER know why a person commits a crime unless you know them very well. If I were going to become a vigilante, then that conclusion would either lead me to: a) start small and do simple things, early Spiderman style. Do things like saving women from being mugged, hanging around dangerous areas out of sight, ready to help the common man at a moment's notice!
Or b) go Batman style: Batman usually didn't fuck around with small fries, he went after the crazies and big time mob bosses that the law wouldn't touch; either because they COULDN'T or because they were corrupt on some level. But that would require some SERIOOOOOUSSSSSSSSSS work! And you'd need to be like a hardcore fucking ninja with Special Forces training and all that. And a millionaire!
So unless you're a ninja millionaire, I'd go with A. Maybe follow some of the small time thugs around until you catch some small time drug dealers or something, who knows, anything would help.
Hmm so how to go about doing that... Well first you DEFINITELY need to make sure NO ONE ever finds out who you are. You couldn't tell a SOUL about it, unless you were ABSOLUTELY sure they would bring your secret with them to the grave. You'd have to work out a way for no one to see your face when you were out around town. You could wear a mask like V in "V for Vendetta," but that would just look silly. I'd take a cue from the Japanese culture and be a ninja. NO ONE EXPECTS A NINJA!
And that brings me to the next issue: training. You'd better be sure you can handle your business out on the streets, or else you're just gonna end up dead and embarrassed for looking like a ninja. Even if you CAN hold your own, you should look into weapons. You don't wanna KILL people, unless of course they DESERVE to die, which is something that's gonna be VERY hard for someone to judge. If you knew someone was let out of jail recently who raped a fucking 4 year old, well I'd say that's a definite candidate for a justice killing. I'd go with an un-traceable sniper rifle with a silencer to just pick'em off from a distance, and a handgun with a silencer for point blank range if POSSIBLE. Otherwise find something similar you can get without being traced. Non lethal weapons I'd go with a tazer for sure, one of those extendo-rod things, brass knuckles, etc... Things that will fuck you up and kill if needed, but designed for non-lethal combat.
Ok so let's see...how about a scenario here? Maybe something more difficult, assuming you couldn't get your hands on something smarter... like that sniper rifle!
Let's say that you were going to go out to kill some child rapist that was STUPIDLY released from jail after like 2 years and now lives in a neighborhood right by some elementary school. You'd have made sure to do your homework, finding out exactly where he lives, his usual business, who comes in and out of his house, if he has a parole officer or FBI agent or whoever keeping tabs on him or watching him. Maybe take a few weeks or so to just study him from a distance and do your research.
When the night comes to take action, you'd pack a backpack full of regular street clothes, hide it somewhere concealed, like a nearby riverbank or bridge or something. Make sure that footprints or some kinda CSI shit wouldn't let anyone trace you to that spot. You'd be wearing a second set of street clothes: a huge black sweater with a hood and big dark jeans. You'd have on your hardcore ninja suit underneath, fully covering your entire body. You'd wait until the time of night with the LEAST amount of people in the streets, and walk to a location a good distance away from the target. Then you would pull your ninja hood over your face and the sweater hood on top of that. That way, no one would be able to see your face but they wouldn't be like "WTF A NINJA!" if they saw you.
You'd run through the shadows or walk with your head down to the target's location (depending on whether or not you could avoid people), make your way quickly to his room, either by scaling and going through a window, or just plain knock on his door and if he opens it, kill him there. Otherwise maybe bust through the door if it's easy enough.
Remember, you're a ninja! No one knows who you are and if they're watching you they can't give a good description of you.
Just make your way quickly to him, kill him with knife/gun/tazer/whatever the hell you're using, and hightail it outta there! Run to the riverbank or wherever you left the backpack with the first set of clothes, change into them, and walk home comfortably. Feign ignorance!
Make sure all parts of your body have been washed and scrubbed and then moisturized so you have as little chance as possible of leaving anything they can get a sample of your DNA or blood from. All parts of your body should be covered. Just to be safe you should burn the suit and clothes afterwards, in case any of his blood or skin or fluids, etc. got on them and you somehow end up being traced to the murder and investigated. Obviously, both the suit and 2 sets of clothes (and shoes) should be bought, thoroughly washed, and kept in plastic covering within 2 days of putting the plan into action, to decrease the chance of anything traceable getting onto the clothes and falling off of you during the murder. Buying the clothes from a second hand store like Salvation Army or Goodwill would be a good choice, as there would be many different DNA samples to pick from, not just yours. You could even look into covering most of your body in liquid latex to be even safer.
Hey, go hard or go home! I'd take as many precautions as possible to make sure I was never caught and never found out. If you SERIOUSLY wanted to become a vigilante, you wouldn't want to just go off and be a dumbass and get into streetfights with thugs hoping that you would come out on top. You'd want to make sure that you could EASILY dispatch them without much difficulty and without killing them, as many times as it takes until you feel your community and the people you care about are safe. That requires desperation and fanatasicm. And if you DO become a vigilante, and you take my advice, and someone traces the killing methods you used to the ones described in my post, they BEST not come and blame me for it!
Haha interesting question though! Good luck to you sir!