This American use of chat up, meaning 'chat with' is new to me, but Jonathan Lighter, author of the Random House Historical Dictionary of American Slang, says that it's been in AmE slang for a few years.
I can think of two possible routes by which this might have come into AmE. One is that it's just another case of putting up after a verb, as in eat up, drink up, call up, smarten up, etc. This up usually has a 'completive' effect--e.g. drink up involves and focus(s)es on the finishing of the drink. It seems unlikely that up has this effect in AmE chat up, since there's no clear point at which chat can be completed--one can always move on to another topic and chat some more. (Incidentally, my students claim that Americans use VERB+up constructions like eat up more than the British do. I think they may just be reacting to the artificiality of the example sentences I use in class, but maybe they're right.)
The other possibility is that this chat up came in some convoluted way from BrE. Here's the scenario I imagine. Some Americans hear British people saying chat up. They think it sounds cool (as British things often sound cool to Americans), so they start saying chat up too. The only problem is that they didn't appreciate what chat up meant in its original context, so they start using it just to mean 'chat with'. This kind of thing happens often when words are borrowed from one language/dialect to another. If you don't know the meaning of a word, there are usually several things in the context that it could mean--so you go for the one that makes most sense to you. If you then start using the word with that misapprehended meaning to people who don' t know that it means something else in the source language, then the word comes to have the new meaning in the target language. In the case of chat up this would be easily done. Here's the scene. Some friends are in a nightclub. One friend rejoins the group and a British friend teases him with:
I saw you chatting up that girlThe meaning 'chatting with' makes perfect sense in this context, but the British friend would have really meant 'I saw you hitting on (or flirting with) that girl.' Hence BrE chat-up line = AmE pick-up line.
Sounds like a good story to me, at least. But very difficult to prove.
I can't mention chat up without mentioning a related piece of BrE, pull, i.e. to get the desired effect of one's chat-up line. The verb pull can be used transitively or intransitively, and it can also be used as a definite noun, especially in the phrase on the pull:
Cilla's not best pleased that she didn't pull a fella when she was away on honeymoon with Yana. She reckons she's got 'married woman' written all over her and that's what's scaring them off. --Corrie Blog
Kareena implores Tariq to come clean about their relationship. She won't let Ash dictate who she can see. He's not keen, so suggests that they keep it quiet for now. She has no idea that he's planning to pull at the party. --Eastenders episode guidenoun
And over in Mike’s apartment, Mike is confused at Leanne being in his home. She is dressed in just a towel and is on her way to have a shower, when Mike calls a friend and asks if he was out on the pull last night. Mike believes that he has had a one night stand with Leanne and makes a move on her. She recoils in horror and tries to explain that she is his son’s girlfriend, but poor Mike is left more confused than ever. --Corrie Blog
In BrE, one can also pull a pint (= 'fill a pint glass with beer/ale from a tap') or pull a face (=AmE make a face). There are a lot of pull phrasal verbs that were originally American--for example pull out, as in Should we pull out of Iraq?. Most of these are used in the UK now, so not so interesting to us here. Have any readers noticed AmE uses of pull that are more mysterious to non-AmE speakers?