The first thing I do when I get to the bar is locate my friends.
These friends might be somebody I already know, or not. Whatever the case, once I've gotten comfortable with this group then they are my "home base." After every approach, every trip to the bar, out to my car to grab something, etc. I return back to this group. I don't quite attach myself to the group the way many people do with their group - moving together. Rather, I come and go, maybe stick around to tell a great story, and as soon as that's done and everybody is still laughing I say "I need to get another drink" or just mutter something and get up... it doesn't matter.
Then if I approach a girl, I always have an excuse first. For example, last one was sitting on the back wall. I said I needed to "sit down for a minute, sober up a bit". This is easy in a small bar, to always have a reason. So I approach a girl like I'm just chatting with her for the moment because she's where I want to be, and when things go especially well I'll start to get summoned by my friends at their table or somebody walking by will say "Hey, so-and-so was looking for you" (especially great if so-and-so is a girl) so then I wind up the conversation, number close, and head back. After that, I pretend not to notice the girl, and if she sees me it's likely at the table with my friends again, at the bar getting a drink, or talking to some other girl. Yes, she can see you talking to other attractive girls (operator being attractive - show her you've got some standards, buddy!), especially with these bar slvts it really doesn't seem to matter.
Just a bit of insight about guys trying not to look like a player at the bar: many girls go to the bar looking to get laid by a guy they know will not trap them in a relationship. I have heard this exact exchange from girls:
girl1: He was cute... but he gave off a player vibe
girl2: great! so he wont be clingy
girl3: yeah, hit it and quit it
Then I started laughing, and they looked at me in shock and said "OMG you weren't supposed to hear that!" and I laughed some more. I guess girls aren't so ashamed of their sexual misdeeds with each other. They just don't want guys to find out.
The tip about calling her a stalker is gold. Running into the girl doesn't have to be a big issue. If you know you're not bringing her home tonight, and you've already settled your business with her by getting the phone number, then you're done with her... for now. Just say something to her teasingly and keep going about your business.