contra bonos mores
There is no disguising the fact that Shadow is an assassin -- everyone around him knows what he does, and he does not go to any great lengths to disguise it. He does not appear to have a home base -- he wanders from town to town, looking for work. He does not limit his work to strictly killing, but he will do just about anything (as long as it involves combat in some way) for the right price. Shadow doesn't seem particularly proud of his occupation, but he doesn't bemoan it, either. He just does it because that's what he knows how to do.
Shadow is not an assassin in the traditional sense of the word. An assassin is usually thought of as someone who murdered a political figure or someone of importance. If we're going to get even more technical, an assassin is the person whose idea it is to murder the political figure -- whether that person murders the political figure is irrelevant. The actual person who kills the political figure (if the person could care less about the implications of the murder and is simply doing it for the money) is a hitman. However, the word assassin can take on the broader meaning of someone who murders for a living, and this is obviously where Shadow fits in. Shadow himself is hardly concerned with politics -- he does what he does for the money.
It is hard to say what drove Shadow to this lifestyle. While he has pretty much always performed sketchy jobs -- formerly he was a thief -- there is a huge difference between stealing and killing. Shadow even had the chance at a peaceful existence in Thamasa, but he chose to turn his back on it an become an assassin. While Shadow does not offer any explanation for his behavior, I think that there were two major factors that contributed to his decision: the (supposed) death of Baram and the death of Relm's mother. Let's take a quick look at Shadow's dream in which he abandons Baram:
Baram: Before you go... You have to use your knife...
Baram: Think what they'll do to me if I get caught. I don't want to go through that. Do me this favor.
Baram: Are you...shaking? I can't believe it! You're acting like a coward! Come on, you weakling! Grab the knife and...
Clyde: I CAN'T!
Clyde: I'm sorry... (Clyde runs away, leaving Baram)
Baram: Clyde! How dare you!?
Shadow is continuously tortured by the fact that he was not able to put his friend out of his misery -- perhaps Baram died a slow death on his own, or perhaps he was brutally tortured by those he stole from -- the fact reminds that Shadow has no idea what happened, and he feels guilty. He was not strong enough to kill his own friend, and it took killing his own emotions and becoming an assassin to gain the strength to be able to do what he couldn't do. In some twisted way, by becoming as assassin, Shadow is better able to cope with what happened to Baram -- if something similar were to happen again, he wouldn't disappoint his friend again.
As for the death of Relm's mother, we never hear firsthand about this event -- we just know that she died sometime before the party meets Relm and Strago, and it can only be assumed that she is the woman in Shadow's flashbacks. It would make sense if Shadow left Thamasa shortly after her death, unable to deal with his grief. The easiest thing for him to do would be to kill his emotions and never look back, and he did just that by becoming an assassin.