To speak to someone with the intent of eventually engaging in sexual intercourse with them. Generally conducted in a relaxed, playful and outstandingly obvious manner.
Almost invariably carried out by men on women; to be a male recipient means you're either extremely lucky or the giver believes you, as the British say, to bat for the other side.
A common, braindead, and therefore quintessentially British phrase, which explains the strange and nonsensical definition(s) sitting next to this one. To chat up is a type of the unfortunate word of banter, which shares all three properties mentioned above.
Guy 1: "Cor, look at all the fit birds in the place!"
Guy 2: "Yeah, I'm gonna chat up the one in the tight black skirt. Phwoar!"
Guy 1 walks in hungover and bruised.
Guy 2: "What did you get up to?"
Guy 1: "I got trashed and started chatting up this stunner. I started pulling and I was all set to bang her but then her boyfriend came back from the bar."
A couple are at a restaurant. The girl goes to the restroom while the guy places the order. When she comes back, the guy and the waitress have just finished talking.
Girl: "I can't take my eyes off you for a second. You were blatantly chatting her up."