This Achievement is easy to get, once you know how it's done, but it does require a dedicated playthrough.
There are moments in the story when the main characters propose alternate options, or when you choose whom to control. When the fight between them happens, the game assigns you the character whose side you have taken the majority of the time. So, basically, if you want the Achievement for playing as Milo after the fight, you need to favour Milo before the fight.
Some of these moments are:
In the Schoolyard Strangler, Lola wants to distract the bouncer; Milo wants to charm Tommy.
To get an invitation to Satan's party, Lola wants to help a demon catch someone; Milo wants to buy a woman a drink.
While buying the woman a drink, one of them has to dance.
When you meet Satan, they both want to do the talking.
When getting Apollyon's Seal, Lola wants to win the trial; Milo wants to lose it.
When getting Asmodeus' Seal, Lola wants to play match maker; Milo just wants to dance.