1. Give them a personality.
Like any character, you need to give your vigilante a well developed personality. I think this becomes even more important for vigilante characters though since they tend to toe the line of morality, you just might need their personality to tip the scales in their favor. If your vigilante is a bland character, our attention will be drawn to his violent actions and we will be less likely to like and therefore sympathize with him in his struggle against evil.
For the brothers in Boondock Saints, this was achieved by balancing their Catholic faith with realistic paradoxes such as their excessive drinking, swearing, and smoking. This was a delicate approach to take that I particularly enjoy. If they were too (traditionally) religious, it might be enough to push them into a terrorist category. As it is, they choose victims based on standard American ethics rather than personal (Catholic) morals. All the same, without their religious affiliation we might question the moral credibility of a couple of Boston Irish men who curse like sailors, smoke like chimneys and drink like fish. I think the MacManus brothers provide an excellent example of the kind of balance you need to achieve in your vigilante characters (particularly if choose to downplay the next component).
2. Give them a push.
Nobody just becomes a vigilante because they feel like it. They have to be desperate enough to occupy the shadowy gap between justice and the law. The Vigilante character must be borne of hardship or depression, either of which can be brought on by the unjust (often violent) death of a close loved-one.
In The Crow, this is the death of both Eric and Shelly. Daredevil- his father. Batman- his parents. The Punisher - Frank's wife and daughter. The Brave One- her fiance. Law Abiding Citizen- his family. V for Vendetta - well, everyone. The only exception to this that I can think of is The Boondock Saints in which case, their reason comes from a supernatural dream. All the same, it came after they learned of a horrible murder (though they themselves were not personally connected to it).
It is usually the trauma and lack of closure and justice for the "bad guys" which pushes them into action.
3. Give them a proper enemy (and a friend).
They say villains are for comic books, not books or movies but when it comes to vigilantes, I disagree. The Antagonist of the piece needs to be established as being quite obviously evil. This doesn't mean they can't be well developed characters, but the reader should never cross over into sympathizing, liking, or (even worse) admiring the villain. The reason this is a huge no-no for the vigilante genre is that when you get to climax - the big fight between your vigilante and the villain, the issue of who is morally superior should never be questioned. If the villain proves to be morally superior (or at least more likable than the hero), when he is arrested (or more often dies) at the hands of the vigilante it becomes unclear whether the act was truly necessary. Which brings to mind the question - was that justice or murder?
Write your villain well but don't lose sight of what you are trying to accomplish. Your villain should be evil not only according to your own religious or cultural assumptions, but in a universal way that probably includes crimes like rape, human trafficking, drug dealing, kidnap, and murder. If the most offensive thing your villain has done is shoplift a candy bar or download a song from the internet, you need to dig deeper and make it grittier.
Another common feature is a friendship that develops between the vigilante and local law enforcement (though it's not always the case, the cop is frequently a minority or otherwise fringe- black, gay, woman, the only one on the force that isn't crooked etc.). In my opinion, this provides the readers with the comfort of knowing that the passionate cop sees the flaws in the legal system and virtually (or actually) gives permission to the vigilante to continue his otherwise questionable activities. In other words, the straight-forward officer (sometimes journalist) that befriends the vigilante essentially provides the reader with the reassurance that the vigilante is "justified". The cop character usually (though not always) tries to get the vigilante to abandon his ways and leave matters in the hands of the professionals. But this doesn't mean they aren't willing to help the vigilante out of sticky situations. If vigilantes were Westerns, the villain would would wear a black hat, the cop white, and the vigilante a gray that changes shades throughout the story.
4. Give them Hell.
If you've set up your story properly (providing a well developed and likable vigilante, a villain that makes your skin crawl, and something to inspire and push your protagonist into action) the next step is setting up "practice" and giving the vigilante a little taste of Hell. Unless he has supernatural powers to aid him (like The Crow) don't make it easy on your vigilante to win. If you choose to balance your drama with humor, provide a few comical errors as the character grapples with the how-to of his new occupation. If you prefer to keep it serious, let a few of the early trials go smoothly and surprise us with a strong opponent and some close calls. Remember, your vigilante probably needs to gain some experience before he works his way up the food chain.
Another aspect to watch for is to make sure neither women nor children are killed (by the vigilante) either on purpose or in the crossfire. This is still a really powerful way to turn your readers against someone which is why it should be reserved for "the bad guys" and not your vigilante.
5. Give them redemption or death.
You can't end your story with the vigilante going to jail. You just can't. He can die. But he can't go to prison -unless he wants to as in Law Abiding Citizen (SPOILER: the film begins with Butler as the vigilante but when he morphs into the villain. Obviously, there's no redemption option here. And so - he dies).
Your other option, like The Brave One, is to end on a positive note in which case the vigilante is emotionally healed (or has worked through their trauma) and no longer feels the need to continue his / her crusade for justice. They can go into retirement, so to speak. As far as I'm aware, those are pretty much your two (traditional) options for an ending for a standalone vigilante piece. If you choose to go with a series, the vigilante can either continue (evil never dies) or can possibly inspire someone to take his place.
Let us know if you have anything to add!