This can be done on either stealth or loud, but doing this stealth will be easier for you. This guide also assumes that you know the basics of the game including key bindings. This achievement is not based on difficulty (it can be played on Normal or Death Sentence: One Down. This achievement also does not require you to complete the heist, if done correctly, it should unlock immediately.Art Gallery is for all attempts and purposes the same as Day 1 of Framing Frame, and if you are familiar with that heist, the same principles apply to the heist here.
Start the heist, stay outside and go to the right hand side of the gallery, near the spiral staircase there should be a door that needs lockpicking (you can also use the alternative methods to enter, but lockpicking is the simplest). This is the external door to the toilet/bathroom where you will be throwing the painting into.
The other side of the room is the internal entrance to the Art Gallery and will immediately be opposite a showroom full of paintings. This heist requires you to steal paintings that are "sold" and are designated as such. This can be done in stealth with not too much difficulty on the lower levels minding the cameras and guards.
Once you have such a painting on your person, go back to the bathroom/toilet and throw the painting on top of one of the open toilets, if done correctly this achievement will unlock, with the downside that the painting will be destroyed and no longer able to be "secured" for profit.
This video by M0rtifer will show you a video demonstration: