First get a high security disguise, head straight to the target, distract her with the grammar-phone, distract or kill the guards in her room she will turn off the grammer-phone and look at the Award you can interact to kill her with this
For next part collect a briefcase these are scattered around the map
For the core turn off the security cameras in the security room on the right hand side of the glass doors leading to the vault get the card from the safe in the room so you can open the vault turn off the laser grid and then put the core in the briefcase, walk to the main entrance exit and just before leaving remove the core form the case then exit and bam the achievement will pop
Video Example:
(Credit to Stev573)