Volume 4 - Chapter 3: Ready for Withdrawal
If Chip died at the end of Volume 3 you will be with Taraneh. After finding out who St John's bank manager (Benedict Hogarth) is through use of the dye, Taraneh will offer to book an appointment with the bank manager. Agree to this and you will go to his office. There you need to convince the bank manager to accommodate you.
Thank you to Fl0ydPinkert0n who has provided the exact chain of dialogue options:How's yours - > Sounds Good - > Oh Fudge, My Bad - > Goodbye to 10 Million
After this the bank manager will accommodate Taraneh/Chip to the vault and the achievement will pop.
However if Chip did not die then he will ask you if he can make the globe rotate as a distraction. Refuse this and you two will go in for an appointment with the bank manager. During this dialogue you need to make the bank manager think that you are rich (I do not remember the exact dialogue options). The final dialogue option should be similar to what it would be with Taraneh. After this the bank manager will accommodate Chip to the vault and the achievement will pop.
If you fail this then immediately reload the checkpoint before the cut scene ends. You will know this as Chip/Taraneh will say something along the lines of "I think that's enough for now, let's go".