You can unlock this achievement as early as Chapter 1. While this minigame is surprisingly easy to complete, you just need.to chat with both girls in order to unlock the achievement. You're not required to be successful with them, though I just needed to chat with both of them once in order to complete all three completion list items.
Anyway, make sure you have a minimum of 6,000 Yen (3,000 per girl) on you, then head over to W. Taihei Blvd. to start the "The Temptation of Live Chat" substory, then head over to Mantai Internet Cafe, where the blue speech bubble is on the map. Once you've completed the substory, talk to the man behind the counter to get started.
Next, purchase 3,030 points (twice if you'd like since you need a total of 6,000 points anyway) and talk to a girl. You have a limited amount of time but it's not hard. Just follow the button commands. Next, chat with the other girl. The achievement unlocks at the end of the second chat.
For reference to make the completion list achievements easier, here are the items you need to complete for this minigame:
Succeed in live chat 10 times
Succeed in live chat 30 times
Complete all girls in live chat
NOTE 1: Succeeding in live chat simply means successfully imputing the button commands and responding to the girl.
NOTE 2: I believe completing a girl means finishing the chat, not necessarily successfully.
NOTE 3: I'm not sure if you actually need to finish the chat in order for it to count.