Living a 'normal life' as Bruce Wayne isn't possible. He's never going to be 'normal' and really, how could he be? He isn't opposed to happiness, but given who he is, it's difficult making it work. His priority is taking back the night.
Whether it's out of character depends on how Nolan portrays it. I don't want to see some garbage where the pressures of being Batman start to wear on him and they pull some crap like they did in Batman Forever. That was out of character. However, if he comes to the realization the he can try to let someone in, it's not that drastic. He's had strong relationships in the past (St. Cloud, Madison, Fairchild, even Talia). The predictable ending was that his standing at Batman compromised all of them. He's not going to sacrifice who he is (Batman) to have a relationship. Selina or Talia work best because they can understand what he goes through better than most. What is comes down to for me is whether this dating thing is something he views can be done or needs to be done. The latter would tick me off.
That response is a little general, because it relates to a movie scenario. If you want to talk specific to a certain medium (TV, comics) let me know.