I agree, cigarette cartons are great items to carry around for this type of playthrough. Selling ammo that you are never going to use is another good way. Also chems like Jet and Psycho will earn a lot if you collect enough, and they weigh nothing.
I sometimes steal if there is something I desperately need. But generally you play it according to the type of character you want to play. Even so, this is all about survival, the need to steal occasionally just to stay alive tends to happen more often than I expected.
I normally read skill books, or store them until I get the Comprehension perk. I don't usually take this perk straight away as there are more useful survival based perks that take priority at the low level stage.
Obviously this is a matter of taste(no pun intended!), but if you have a character with the Cannibal perk you can earn a lot of caps selling your Stimpacks(or at least the majority of them). You may consider this to be cheating (I don't, as they changed the Level it is given in NV), but I use the console to give myself the Cannibal Perk as soon as I leave the Vault. It's not like it gives you a huge advantage having it(as there are a lot of negative aspects to it of course), I just find that it adds to the survival-at-any-cost roleplaying aspect.
I added a little bit of Role-Playing to it. My current character(my only Cannibal character so far), Daphne, wandered the wastes for a while and decided to explore rather than go to Megaton right away. She got lost and severely wounded by Raiders. However, she was able to kill the Raiders but had no place to rest. She was also starving and couldn't go any longer without food. That was the day she "officially" became a Cannibal. These events did happen, I just added the starvation part to it.
I am just surprised that Jericho, even as evil as he is, hasn't made mention of my eating habits. :blush2: