Like a Dragon Gaiden: The Man Who Erased His Name

You can improve your bond with Akame by going through the Drink Link events (sort of similar to how it was done between Ichiban and his party members in his game) that are unlocked by bringing the Akame Network to lvl 13, 16 and 19. However, even if you achieve those levels, the events won't trigger until you've reached the point of the story where you have obtained the Gold Castle rank (Akame Network at lvl 10 and 3 Silver fights completed in the Coliseum). After Akame gives you the Gold Tiger that symbolizes your rank-up, you will have the opportunity to trigger these events when you talk to her.

The Drink Link events will give you some backstory on Akame and her past, with some dialogue choices here and there (the choices don't appear to have any impact so answer the way you like) and you'll then be rewarded with the chance to invite Akame to play with you at darts, Pocket Circuit and karaoke. The last event will include a fight against some thugs and the achievement should pop during/after the debrief that concludes this event.