Easy achievement, head to the far west part of "City of Westminster" district (if you zoom in enough the area will glow red and you can see the name Buckingham Palace) save one of those guys in red uniforms known as the "Royal Guards" to your recruits. complete the recruit mission and simply switch to them and walk in the Palace while wearing the uniform.
You need to buy the Deep Profile skill using the tech pointsCredit to Karma Arsonist
For the above: you may have to use the deep profile option if you are running into issues finding one outside the palace itself.
Do the deep profile > pick one of the recruitment missions it tells you about for the person you deep profiled > you should get the person after that > deal to the person and go into the palace and done.
NewbieKenobi mentioned If you can't get this to unlock using your Royal Guard, go in from the big entrance facing a large fountain on the north east side of the palace