6'0 120 lbs doesn't sound too good. I have a friend who is 5'6 and 120 and he's like the skinniest guy I know, so I can't imagine another 6 inches with no extra weight. I've always tried to gain weight and had trouble doing it, but I've managed to gain close to 10 pounds since college started. So now I'm around 200 lbs, which is a good weight for me I guess. It's all about just finding a weight at which you feel healthy. I don't really know what my recommended weight is, since most charts on the subject stop at around 6'5, but I'd say 200-210 is probably a safe bet.
I found an online chart just now and it says that a 6'0 male should be anywhere from 149-188 depending on frame size. I don't know how much trust to put in these things though, so if you don't feel a need to worry about it, it shouldn't be a problem.