It's not as complicated as it sounds. From what I can tell, you just need to win the Koharu mini-game three times. It becomes available after you have played a few of the story missions (TBD: need to add info on exactly when it becomes available). It will lock after beating it each time. It will unlock again after you spend time with Natalie (or Betty, if you have the DLC installed).
Use money earned in missions to buy gifts in the "Office" menu. Like Natalia, Koharu likes flowers. Also, the fan. Other gifts she likes can be seen when you obtain the nurse's x-ray glasses. (TBD: Add a list of gifts Koharu likes). This can be brute-forced pretty easily just by doing the mission well-stocked with gifts and trying different ones.
The achievement unlocks after the third time. From what I noticed, the ending cutscene changes/extends a little each time. And after the first time, the weird "I'm gonna get you" hands just before he spins her around has to be one of the creepiest, stalker-like things I've ever seen in a game.
It's actually pretty easy to power through this without worrying too much about playing it the way you're supposed to (which is to say, being cautions, looking her in the eyes, and so forth). The trick is that moving the right stick to the right "looks away" and restores the meter on the right. So just watch the vertical meter on the right-hand side. Move the right stick down and hold LT and the "blood" in the head will keep filling. Also watch the meter on the right. It doesn't matter if it's "buzzing" because you're being caught looking. As it gets under half way empty, but not all the way (this will probably happen very fast), let off LT and and move the right stick right until it fills that meter back up. Then repeat. Once the "blood" hits 1000 you can offer a gift, and it's easy to do this quite a few times even offering "bad" gifts and still win the mission. Seems to work every time.
The game gives an explanation on how to play the mini-game, but basically you hold LT for "Gigolo Vision". Look at her eyes when she's watching you. The moment she turns away, move the stick down and hold LT (to ogle her) until you get a flashing/tone indicating a score increase. The "blood" in the head on the left will fill up each time. Keep looking up and go back to her eyes as soon as she's looking again. When the blood reaches 1000, you can hold RT to offer a gift. Occassionally, let off LT and move the stick to the right to "look away" for a few seconds. That's pretty much all there is to it. Yes, it's as creepy and uncomfortable as it sounds.