As long as an enemy does not have any grey lightning bolts above its head, it is unaware. Attacking them with your sword at any time they are unaware will result in an automatic assassination. The assassination will look much different than a regular sword kill because it will have a unique, longer and much more gruesome animation sequence.
You can also assassinate a target by performing a Drop Assassination—5 of which are needed for the Razor Rain trophy. To perform a Drop Assassination you must drop on an enemy from above and press just before landing on them. You will always see the button prompt come up as soon as you are within range of performing the assassination.
The third and final way to assassinate a target is by using Bend Time 2. Any and all attacks performed during Bend Time level 2 will automatically result in an assassination, but remember: they must be unaware in order to count towards the trophy.
This is an extremely easy trophy to obtain and will most likely unlock naturally during your Dunwall in Chaos playthrough without even trying.