What you are talking about is called a Dark Protagonist.
Yes. This is possible, if handled correctly. If handled incorrectly, your main character will fail and take the book with her.
The trick with any protagonist (and indeed any part of writing) is to consider what you want for the reader. The main thing that you want is for the reader to keep reading, all the way through your book. A protagonist that he doesn't sympathize with (or worse, doesn't like) is going to seriously detract from his experience, and possibly cause him to stop reading. Creating dark protagonists is tricky, especially for new writers. However, it can be done.
If you're a new writer, my first answer would be to not write a dark protagonist. Seriously consider if your protagonist can be observing these qualities in someone else. That will really help.
However, being a writer myself, I know these ideas have a way of sticking with us, so if you want to write a dark protagonist, there is a very particular way you need to go about it.
No one wants to read about a purely evil person. I'm sorry, that's just the way it is. If your protagonist is a murderer or even something as simple as a thief, you're in trouble. In order to fix this problem, you have to think about what a reader does want to read.
In terms of characters, readers want to read about heroes. You might be tired of heroes from reading about too many poorly developed protagonists, but hear me out. A reader wants to read about someone that they can look up to. They want to see the qualities that they themselves wish they had. They want to read about people who are willing to do the things or say the words that they would never dare, but would nonetheless applaud. A character that is this way is called larger than life.
So how does this apply to dark protagonists? There really isn't much difference. The dark protagonist still needs something - some quality or mentality - that makes her heroic in some way. And when it comes to dark protagonists, this usually comes out as a sign that the protagonist is not entirely dark.
@Stephen used Jason Bourne as an example, so I'll use him too. Bourne kills ruthlessly, and appears to do so with no emotion whatsoever. How on earth are we behind him? As Stephen pointed out, it's the reasoning behind what he does. We empathize with his mission (which come from a different part of character development), and so sympathize with him and whatever it takes to complete that mission. And we discover that he has emotions, cares for people, and only kills when necessary (think about it: he never kills unless it's a trained assassin that actually poses a threat to him).
Another good example is Artemis Fowl. I don't know if you've read the Artemis Fowl books, but they are about a young boy who is a criminal mastermind, stealing simply to increase his own wealth. Worse, he keeps his ill mother in the dark about what he's doing, and kidnaps a certain 'gifted' individual simply to learn her powers. How does the novel work? Artemis is not dark. Not entirely, anyway. He's devious and evil, certainly, but he cannot suppress the feelings within him. He cares for his mother. He sympathizes with his prisoner. While he feels a thrill during a robbery, he even feels sorry for that afterwards.
So how do you create a workable dark protagonist? Show that she is not entirely dark. Give the reader a hint, a glimmer of hope, something that they can cheer for whenever it wins over the protagonist's darkness. I doubt your story is about how the actions of your protagonist are right. Give her doubts. She can suppress them all she wants; they'll still be there. As long as they are, and as long as they refuse to go away, resurfacing at every opportunity, the reader has hope. Keep that hope alive, and your protagonist will succeed.
Good luck in your writing!